Properties

Label 4-1632e2-1.1-c1e2-0-41
Degree $4$
Conductor $2663424$
Sign $-1$
Analytic cond. $169.822$
Root an. cond. $3.60992$
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  + 9-s − 2·19-s + 25-s − 16·41-s + 22·43-s − 2·49-s + 16·59-s − 16·67-s − 8·73-s + 81-s − 12·83-s − 12·89-s − 8·97-s + 16·107-s − 21·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 15·169-s − 2·171-s + 173-s + ⋯
L(s)  = 1  + 1/3·9-s − 0.458·19-s + 1/5·25-s − 2.49·41-s + 3.35·43-s − 2/7·49-s + 2.08·59-s − 1.95·67-s − 0.936·73-s + 1/9·81-s − 1.31·83-s − 1.27·89-s − 0.812·97-s + 1.54·107-s − 1.90·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.15·169-s − 0.152·171-s + 0.0760·173-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2663424 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2663424 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2663424\)    =    \(2^{10} \cdot 3^{2} \cdot 17^{2}\)
Sign: $-1$
Analytic conductor: \(169.822\)
Root analytic conductor: \(3.60992\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{2663424} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 2663424,\ (\ :1/2, 1/2),\ -1)\)

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 \( 1 \)
3$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
17$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good5$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \)
7$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
13$C_2^2$ \( 1 - 15 T^{2} + p^{2} T^{4} \)
19$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
23$C_2^2$ \( 1 + 3 T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
31$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \)
37$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \)
41$C_2$$\times$$C_2$ \( ( 1 + 5 T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 11 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 + 42 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \)
59$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - 6 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
67$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
71$C_2^2$ \( 1 - 110 T^{2} + p^{2} T^{4} \)
73$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
79$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \)
83$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
89$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \)
97$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
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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

−7.32669361309404544253638892775, −7.12308454345833641207141807395, −6.51176431234695390752480750643, −6.25657820354822724690977541288, −5.59925972547563226148263711683, −5.41747783371230540887411391057, −4.79694676992625594175431971013, −4.23982348710988972447801603144, −4.06675786273926021118743637775, −3.39628535070214125817038623917, −2.82255866418502889752956409955, −2.37093323908239350072883358473, −1.66410651305856432119264520711, −1.05470376963053088734525219747, 0, 1.05470376963053088734525219747, 1.66410651305856432119264520711, 2.37093323908239350072883358473, 2.82255866418502889752956409955, 3.39628535070214125817038623917, 4.06675786273926021118743637775, 4.23982348710988972447801603144, 4.79694676992625594175431971013, 5.41747783371230540887411391057, 5.59925972547563226148263711683, 6.25657820354822724690977541288, 6.51176431234695390752480750643, 7.12308454345833641207141807395, 7.32669361309404544253638892775

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