Properties

Degree 2
Conductor $ 2^{2} \cdot 3 \cdot 13 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 2·7-s + 9-s + 13-s − 6·17-s + 2·19-s + 2·21-s − 5·25-s + 27-s − 6·29-s + 2·31-s + 2·37-s + 39-s − 12·41-s − 4·43-s − 3·49-s − 6·51-s + 6·53-s + 2·57-s + 12·59-s + 2·61-s + 2·63-s − 10·67-s + 12·71-s + 14·73-s − 5·75-s + 8·79-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.755·7-s + 1/3·9-s + 0.277·13-s − 1.45·17-s + 0.458·19-s + 0.436·21-s − 25-s + 0.192·27-s − 1.11·29-s + 0.359·31-s + 0.328·37-s + 0.160·39-s − 1.87·41-s − 0.609·43-s − 3/7·49-s − 0.840·51-s + 0.824·53-s + 0.264·57-s + 1.56·59-s + 0.256·61-s + 0.251·63-s − 1.22·67-s + 1.42·71-s + 1.63·73-s − 0.577·75-s + 0.900·79-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(156\)    =    \(2^{2} \cdot 3 \cdot 13\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{156} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 156,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.376149593$
$L(\frac12)$  $\approx$  $1.376149593$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;13\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;13\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 \)
3 \( 1 - T \)
13 \( 1 - T \)
good5 \( 1 + p T^{2} \)
7 \( 1 - 2 T + p T^{2} \)
11 \( 1 + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 12 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 10 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−19.40547320749622, −18.32955031914811, −17.71757871284770, −16.63956165839831, −15.50188387831543, −14.92621776545525, −13.78414402825761, −13.21941002116463, −11.83258307695703, −11.02368824085057, −9.798663464890124, −8.735814680045936, −7.868716408824418, −6.658438911776524, −5.114545706985201, −3.800678541076204, −2.038817583621088, 2.038817583621088, 3.800678541076204, 5.114545706985201, 6.658438911776524, 7.868716408824418, 8.735814680045936, 9.798663464890124, 11.02368824085057, 11.83258307695703, 13.21941002116463, 13.78414402825761, 14.92621776545525, 15.50188387831543, 16.63956165839831, 17.71757871284770, 18.32955031914811, 19.40547320749622

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