Properties

Degree 4
Conductor $ 2^{2} \cdot 7^{2} \cdot 67 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 3·4-s − 7-s − 4·8-s + 9-s + 2·14-s + 5·16-s − 2·18-s + 9·23-s + 8·25-s − 3·28-s − 6·32-s + 3·36-s + 4·37-s + 7·43-s − 18·46-s − 6·49-s − 16·50-s − 18·53-s + 4·56-s − 63-s + 7·64-s + 15·67-s + 3·71-s − 4·72-s − 8·74-s − 2·79-s + ⋯
L(s)  = 1  − 1.41·2-s + 3/2·4-s − 0.377·7-s − 1.41·8-s + 1/3·9-s + 0.534·14-s + 5/4·16-s − 0.471·18-s + 1.87·23-s + 8/5·25-s − 0.566·28-s − 1.06·32-s + 1/2·36-s + 0.657·37-s + 1.06·43-s − 2.65·46-s − 6/7·49-s − 2.26·50-s − 2.47·53-s + 0.534·56-s − 0.125·63-s + 7/8·64-s + 1.83·67-s + 0.356·71-s − 0.471·72-s − 0.929·74-s − 0.225·79-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 13132 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 13132 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(13132\)    =    \(2^{2} \cdot 7^{2} \cdot 67\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{13132} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 13132,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $0.5711183854$
$L(\frac12)$  $\approx$  $0.5711183854$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;7,\;67\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;7,\;67\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad2$C_1$ \( ( 1 + T )^{2} \)
7$C_2$ \( 1 + T + p T^{2} \)
67$C_1$$\times$$C_2$ \( ( 1 - T )( 1 - 14 T + p T^{2} ) \)
good3$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \)
5$C_2^2$ \( 1 - 8 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
13$C_2^2$ \( 1 - 8 T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
19$C_2^2$ \( 1 - 20 T^{2} + p^{2} T^{4} \)
23$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 - 3 T + p T^{2} ) \)
29$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
31$C_2^2$ \( 1 + 19 T^{2} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 - 8 T^{2} + p^{2} T^{4} \)
43$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + T + p T^{2} ) \)
47$C_2^2$ \( 1 + 40 T^{2} + p^{2} T^{4} \)
53$C_2$$\times$$C_2$ \( ( 1 + 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
61$C_2^2$ \( 1 - 86 T^{2} + p^{2} T^{4} \)
71$C_2$$\times$$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
73$C_2^2$ \( 1 - 29 T^{2} + p^{2} T^{4} \)
79$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
83$C_2^2$ \( 1 - 32 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 - 83 T^{2} + p^{2} T^{4} \)
97$C_2^2$ \( 1 + 148 T^{2} + p^{2} T^{4} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−11.06242556734797633316617654325, −10.75142325686499068465966747941, −10.03580089431797300060009499074, −9.496854124180573508165716157921, −9.145978517862082072816079858900, −8.570611933157460167149885055115, −7.946093092912831285360621515242, −7.34391638071145319827988985542, −6.73529519082847451308985885003, −6.38183167189614169029600475754, −5.35998631133052630855741203130, −4.60746508936177210269135914867, −3.34534227552155671660153330050, −2.62780247878860436110860336078, −1.19064202541676050214281881703, 1.19064202541676050214281881703, 2.62780247878860436110860336078, 3.34534227552155671660153330050, 4.60746508936177210269135914867, 5.35998631133052630855741203130, 6.38183167189614169029600475754, 6.73529519082847451308985885003, 7.34391638071145319827988985542, 7.946093092912831285360621515242, 8.570611933157460167149885055115, 9.145978517862082072816079858900, 9.496854124180573508165716157921, 10.03580089431797300060009499074, 10.75142325686499068465966747941, 11.06242556734797633316617654325

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