Properties

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

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s − 2·5-s − 2·7-s + 9-s − 6·11-s + 2·13-s + 4·15-s + 17-s + 4·21-s + 6·23-s − 25-s + 4·27-s − 10·29-s + 2·31-s + 12·33-s + 4·35-s + 6·37-s − 4·39-s − 6·41-s − 8·43-s − 2·45-s − 3·49-s − 2·51-s − 10·53-s + 12·55-s − 8·59-s + 14·61-s + ⋯
L(s)  = 1  − 1.15·3-s − 0.894·5-s − 0.755·7-s + 1/3·9-s − 1.80·11-s + 0.554·13-s + 1.03·15-s + 0.242·17-s + 0.872·21-s + 1.25·23-s − 1/5·25-s + 0.769·27-s − 1.85·29-s + 0.359·31-s + 2.08·33-s + 0.676·35-s + 0.986·37-s − 0.640·39-s − 0.937·41-s − 1.21·43-s − 0.298·45-s − 3/7·49-s − 0.280·51-s − 1.37·53-s + 1.61·55-s − 1.04·59-s + 1.79·61-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  =  1
Selberg data  =  $(2,\ 136,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   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 \( 1 + 2 T + p T^{2} \)
7 \( 1 + 2 T + p T^{2} \)
11 \( 1 + 6 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 10 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 2 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 - 8 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−18.88752225483208, −18.49906292472557, −17.33040335584030, −16.40901819969467, −15.83176919906966, −15.00013464085988, −13.27041684568587, −12.72735841786322, −11.50681162497893, −10.94256231134113, −9.867199747238314, −8.306702051055435, −7.229596471732923, −5.947662974655939, −4.939332209834613, −3.247437361825062, 0, 3.247437361825062, 4.939332209834613, 5.947662974655939, 7.229596471732923, 8.306702051055435, 9.867199747238314, 10.94256231134113, 11.50681162497893, 12.72735841786322, 13.27041684568587, 15.00013464085988, 15.83176919906966, 16.40901819969467, 17.33040335584030, 18.49906292472557, 18.88752225483208

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