Properties

Degree 2
Conductor $ 2^{3} \cdot 17 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s + 9-s + 2·11-s − 6·13-s − 17-s + 4·19-s + 4·23-s − 5·25-s − 4·27-s − 8·31-s + 4·33-s − 4·37-s − 12·39-s + 6·41-s + 8·43-s − 8·47-s − 7·49-s − 2·51-s + 10·53-s + 8·57-s + 12·61-s + 8·67-s + 8·69-s + 12·71-s + 2·73-s − 10·75-s − 4·79-s + ⋯
L(s)  = 1  + 1.15·3-s + 1/3·9-s + 0.603·11-s − 1.66·13-s − 0.242·17-s + 0.917·19-s + 0.834·23-s − 25-s − 0.769·27-s − 1.43·31-s + 0.696·33-s − 0.657·37-s − 1.92·39-s + 0.937·41-s + 1.21·43-s − 1.16·47-s − 49-s − 0.280·51-s + 1.37·53-s + 1.05·57-s + 1.53·61-s + 0.977·67-s + 0.963·69-s + 1.42·71-s + 0.234·73-s − 1.15·75-s − 0.450·79-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(136\)    =    \(2^{3} \cdot 17\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{136} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 136,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.433630837$
$L(\frac12)$  $\approx$  $1.433630837$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;17\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;17\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 \)
17 \( 1 + T \)
good3 \( 1 - 2 T + p T^{2} \)
5 \( 1 + p T^{2} \)
7 \( 1 + p T^{2} \)
11 \( 1 - 2 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 12 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 + 18 T + p T^{2} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−19.61348467141781, −19.20007983043585, −17.84848505220000, −17.04492618385585, −15.92232007758380, −14.76487932002052, −14.41851015966023, −13.37507819601213, −12.31736659192486, −11.24949424038559, −9.716492599218372, −9.181675232501276, −7.915799644978553, −7.040288092457812, −5.277404120646219, −3.698930876354664, −2.331452969982914, 2.331452969982914, 3.698930876354664, 5.277404120646219, 7.040288092457812, 7.915799644978553, 9.181675232501276, 9.716492599218372, 11.24949424038559, 12.31736659192486, 13.37507819601213, 14.41851015966023, 14.76487932002052, 15.92232007758380, 17.04492618385585, 17.84848505220000, 19.20007983043585, 19.61348467141781

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