Properties

Label 4-758912-1.1-c1e2-0-4
Degree $4$
Conductor $758912$
Sign $1$
Analytic cond. $48.3888$
Root an. cond. $2.63746$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s − 8-s − 2·9-s + 2·11-s + 16-s + 2·18-s − 2·22-s − 10·25-s − 32-s − 2·36-s + 8·43-s + 2·44-s − 7·49-s + 10·50-s + 64-s + 8·67-s + 2·72-s − 5·81-s − 8·86-s − 2·88-s + 7·98-s − 4·99-s − 10·100-s + 24·107-s − 4·113-s + 3·121-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.353·8-s − 2/3·9-s + 0.603·11-s + 1/4·16-s + 0.471·18-s − 0.426·22-s − 2·25-s − 0.176·32-s − 1/3·36-s + 1.21·43-s + 0.301·44-s − 49-s + 1.41·50-s + 1/8·64-s + 0.977·67-s + 0.235·72-s − 5/9·81-s − 0.862·86-s − 0.213·88-s + 0.707·98-s − 0.402·99-s − 100-s + 2.32·107-s − 0.376·113-s + 3/11·121-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 758912 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 758912 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(758912\)    =    \(2^{7} \cdot 7^{2} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(48.3888\)
Root analytic conductor: \(2.63746\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{758912} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 758912,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.022750326\)
\(L(\frac12)\) \(\approx\) \(1.022750326\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_1$ \( 1 + T \)
7$C_2$ \( 1 + p T^{2} \)
11$C_1$ \( ( 1 - T )^{2} \)
good3$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
5$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
31$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
41$C_2^2$ \( 1 - 74 T^{2} + p^{2} T^{4} \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 + 86 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
59$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2^2$ \( 1 + 54 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
83$C_2^2$ \( 1 + 122 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.210782209230272924399352563142, −7.927526419773322101411816402136, −7.50968336511021633948791790295, −6.99358558307192799019139977645, −6.53061291350032711609374346167, −5.97809508402494877786036277002, −5.79145013083096397789137362278, −5.18107078182225530920591517213, −4.50305841090105781069805565787, −3.95408726760756327519159600691, −3.47234161647353864334980484708, −2.82474102198788392759886303315, −2.15763197755219219838506453000, −1.59402090028731607000500257794, −0.54968632138568345348621500265, 0.54968632138568345348621500265, 1.59402090028731607000500257794, 2.15763197755219219838506453000, 2.82474102198788392759886303315, 3.47234161647353864334980484708, 3.95408726760756327519159600691, 4.50305841090105781069805565787, 5.18107078182225530920591517213, 5.79145013083096397789137362278, 5.97809508402494877786036277002, 6.53061291350032711609374346167, 6.99358558307192799019139977645, 7.50968336511021633948791790295, 7.927526419773322101411816402136, 8.210782209230272924399352563142

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