Properties

Label 4-1148e2-1.1-c1e2-0-7
Degree $4$
Conductor $1317904$
Sign $-1$
Analytic cond. $84.0307$
Root an. cond. $3.02767$
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-s − 4-s + 4·5-s − 3·8-s − 3·9-s + 4·10-s − 16-s − 4·17-s − 3·18-s − 4·20-s + 3·25-s + 14·29-s + 5·32-s − 4·34-s + 3·36-s − 4·37-s − 12·40-s − 12·41-s − 12·45-s + 49-s + 3·50-s − 2·53-s + 14·58-s + 4·61-s + 7·64-s + 4·68-s + 9·72-s + ⋯
L(s)  = 1  + 0.707·2-s − 1/2·4-s + 1.78·5-s − 1.06·8-s − 9-s + 1.26·10-s − 1/4·16-s − 0.970·17-s − 0.707·18-s − 0.894·20-s + 3/5·25-s + 2.59·29-s + 0.883·32-s − 0.685·34-s + 1/2·36-s − 0.657·37-s − 1.89·40-s − 1.87·41-s − 1.78·45-s + 1/7·49-s + 0.424·50-s − 0.274·53-s + 1.83·58-s + 0.512·61-s + 7/8·64-s + 0.485·68-s + 1.06·72-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1317904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1317904 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1317904\)    =    \(2^{4} \cdot 7^{2} \cdot 41^{2}\)
Sign: $-1$
Analytic conductor: \(84.0307\)
Root analytic conductor: \(3.02767\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1317904} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 1317904,\ (\ :1/2, 1/2),\ -1)\)

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$C_2$ \( 1 - T + p T^{2} \)
7$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
41$C_2$ \( 1 + 12 T + p T^{2} \)
good3$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
5$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 - T + p T^{2} ) \)
11$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
17$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \)
19$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
23$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
29$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 - 6 T + p T^{2} ) \)
31$C_2^2$ \( 1 - 27 T^{2} + p^{2} T^{4} \)
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 + 55 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 39 T^{2} + p^{2} T^{4} \)
53$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2^2$ \( 1 + 115 T^{2} + p^{2} T^{4} \)
61$C_2$$\times$$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
67$C_2^2$ \( 1 - 97 T^{2} + p^{2} T^{4} \)
71$C_2^2$ \( 1 + 73 T^{2} + p^{2} T^{4} \)
73$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 - 4 T + p T^{2} ) \)
79$C_2^2$ \( 1 + 89 T^{2} + p^{2} T^{4} \)
83$C_2^2$ \( 1 + 91 T^{2} + p^{2} T^{4} \)
89$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
97$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
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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

−7.912557315676358567794666403783, −7.07252118152093106605624065275, −6.60200851238385647996145024794, −6.31951729401620381379585261696, −6.02336898838701487042227128106, −5.45291163293155166850096601088, −5.13548703344812413123160320253, −4.82405407040722146505324502341, −4.19728318460680687523126437985, −3.59568732569069429908453888433, −2.98071006984466882116716931515, −2.54351357308048197224785892344, −2.05715121046755611720399459632, −1.21189548615431144984756203024, 0, 1.21189548615431144984756203024, 2.05715121046755611720399459632, 2.54351357308048197224785892344, 2.98071006984466882116716931515, 3.59568732569069429908453888433, 4.19728318460680687523126437985, 4.82405407040722146505324502341, 5.13548703344812413123160320253, 5.45291163293155166850096601088, 6.02336898838701487042227128106, 6.31951729401620381379585261696, 6.60200851238385647996145024794, 7.07252118152093106605624065275, 7.912557315676358567794666403783

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