Properties

Label 4-901404-1.1-c1e2-0-7
Degree $4$
Conductor $901404$
Sign $-1$
Analytic cond. $57.4743$
Root an. cond. $2.75339$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s − 4-s − 7-s + 9-s − 2·12-s − 2·13-s + 16-s − 2·21-s + 2·25-s − 4·27-s + 28-s − 8·31-s − 36-s + 16·37-s − 4·39-s − 14·43-s + 2·48-s + 49-s + 2·52-s + 12·61-s − 63-s − 64-s − 73-s + 4·75-s − 4·79-s − 11·81-s + 2·84-s + ⋯
L(s)  = 1  + 1.15·3-s − 1/2·4-s − 0.377·7-s + 1/3·9-s − 0.577·12-s − 0.554·13-s + 1/4·16-s − 0.436·21-s + 2/5·25-s − 0.769·27-s + 0.188·28-s − 1.43·31-s − 1/6·36-s + 2.63·37-s − 0.640·39-s − 2.13·43-s + 0.288·48-s + 1/7·49-s + 0.277·52-s + 1.53·61-s − 0.125·63-s − 1/8·64-s − 0.117·73-s + 0.461·75-s − 0.450·79-s − 1.22·81-s + 0.218·84-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 901404 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 901404 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(901404\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{3} \cdot 73\)
Sign: $-1$
Analytic conductor: \(57.4743\)
Root analytic conductor: \(2.75339\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{901404} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 901404,\ (\ :1/2, 1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T^{2} \)
3$C_2$ \( 1 - 2 T + p T^{2} \)
7$C_1$ \( 1 + T \)
73$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + 2 T + p T^{2} ) \)
good5$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
13$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \)
17$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
23$C_2^2$ \( 1 + 22 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
31$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \)
43$C_2$$\times$$C_2$ \( ( 1 + 6 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
47$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 62 T^{2} + p^{2} T^{4} \)
61$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
71$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
79$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
83$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
97$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
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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

−8.101451523332059389441414220934, −7.53012283520830470672949513330, −7.24678266930791089497711344659, −6.78209539771573183653717009592, −6.04087840134255496261575183416, −5.85922803970098985033481789005, −4.98869808527250273453434285248, −4.87339579371396446561727960108, −3.99236403007372923060728658470, −3.75080368839514761585140542836, −3.14472260752924686000824118892, −2.60651825696654980537491098183, −2.13082877707974009998466704081, −1.21766576142469790564065907187, 0, 1.21766576142469790564065907187, 2.13082877707974009998466704081, 2.60651825696654980537491098183, 3.14472260752924686000824118892, 3.75080368839514761585140542836, 3.99236403007372923060728658470, 4.87339579371396446561727960108, 4.98869808527250273453434285248, 5.85922803970098985033481789005, 6.04087840134255496261575183416, 6.78209539771573183653717009592, 7.24678266930791089497711344659, 7.53012283520830470672949513330, 8.101451523332059389441414220934

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