Properties

Label 24-371e12-1.1-c0e12-0-1
Degree $24$
Conductor $6.800\times 10^{30}$
Sign $1$
Analytic cond. $1.62320\times 10^{-9}$
Root an. cond. $0.430294$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 4-s − 7-s + 9-s + 2·11-s + 25-s + 28-s − 2·29-s − 36-s + 2·37-s − 11·43-s − 2·44-s − 53-s − 63-s − 2·77-s − 13·79-s + 2·99-s − 100-s + 2·107-s − 2·113-s + 2·116-s + 121-s + 127-s + 131-s + 137-s + 139-s − 2·148-s + 149-s + ⋯
L(s)  = 1  − 4-s − 7-s + 9-s + 2·11-s + 25-s + 28-s − 2·29-s − 36-s + 2·37-s − 11·43-s − 2·44-s − 53-s − 63-s − 2·77-s − 13·79-s + 2·99-s − 100-s + 2·107-s − 2·113-s + 2·116-s + 121-s + 127-s + 131-s + 137-s + 139-s − 2·148-s + 149-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{12} \cdot 53^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{12} \cdot 53^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(24\)
Conductor: \(7^{12} \cdot 53^{12}\)
Sign: $1$
Analytic conductor: \(1.62320\times 10^{-9}\)
Root analytic conductor: \(0.430294\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{371} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((24,\ 7^{12} \cdot 53^{12} ,\ ( \ : [0]^{12} ),\ 1 )\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.06619052538\)
\(L(\frac12)\) \(\approx\) \(0.06619052538\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} \)
53 \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} \)
good2 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} ) \)
3 \( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} - T^{22} + T^{24} \)
5 \( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} - T^{22} + T^{24} \)
11 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12} )^{2} \)
13 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} ) \)
17 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} ) \)
19 \( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} - T^{22} + T^{24} \)
23 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} ) \)
29 \( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} )^{2} \)
31 \( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} - T^{22} + T^{24} \)
37 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12} )^{2} \)
41 \( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} - T^{22} + T^{24} \)
43 \( ( 1 + T )^{12}( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12} ) \)
47 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} ) \)
59 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} ) \)
61 \( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} - T^{22} + T^{24} \)
67 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} ) \)
71 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} ) \)
73 \( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} - T^{14} + T^{16} - T^{18} + T^{20} - T^{22} + T^{24} \)
79 \( ( 1 + T )^{12}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} ) \)
83 \( ( 1 + T^{2} )^{12} \)
89 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} ) \)
97 \( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} - T^{11} + T^{12} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} + T^{11} + T^{12} ) \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−4.23348208975434856150239876541, −4.22096651321361481317421138949, −3.93041876209085254450911570499, −3.83870131587512907052877141328, −3.79430557048283689872937116494, −3.66531791154729953407607735238, −3.50796112988515057911566852065, −3.24450743741590888238517228522, −3.23420552561449385488935081088, −3.14640688255150878889122022921, −3.10624843344201519467595704101, −3.07534098462048307307805656613, −3.02336718408721931988023836550, −2.71045486592247094564805590063, −2.68538716729119210150901255493, −2.56436017484390887005928208433, −2.31641175969030792232293481345, −1.92311209447126202780843368995, −1.81414587305780611289115545351, −1.69596574132579481149936483448, −1.66084265745210498443051709844, −1.55790800907781535015309252584, −1.55722730916122140494014399858, −1.36465365339080332263066607877, −1.05999909066573729198186995302, 1.05999909066573729198186995302, 1.36465365339080332263066607877, 1.55722730916122140494014399858, 1.55790800907781535015309252584, 1.66084265745210498443051709844, 1.69596574132579481149936483448, 1.81414587305780611289115545351, 1.92311209447126202780843368995, 2.31641175969030792232293481345, 2.56436017484390887005928208433, 2.68538716729119210150901255493, 2.71045486592247094564805590063, 3.02336718408721931988023836550, 3.07534098462048307307805656613, 3.10624843344201519467595704101, 3.14640688255150878889122022921, 3.23420552561449385488935081088, 3.24450743741590888238517228522, 3.50796112988515057911566852065, 3.66531791154729953407607735238, 3.79430557048283689872937116494, 3.83870131587512907052877141328, 3.93041876209085254450911570499, 4.22096651321361481317421138949, 4.23348208975434856150239876541

Graph of the $Z$-function along the critical line

Plot not available for L-functions of degree greater than 10.