Properties

Degree 2
Conductor 31
Sign $1$
Motivic weight 0
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 5-s − 7-s + 8-s + 9-s + 10-s + 14-s − 16-s − 18-s − 19-s + 31-s + 35-s + 38-s − 40-s − 41-s − 45-s + 2·47-s − 56-s − 59-s − 62-s − 63-s + 64-s + 2·67-s − 70-s − 71-s + 72-s + 80-s + ⋯
L(s)  = 1  − 2-s − 5-s − 7-s + 8-s + 9-s + 10-s + 14-s − 16-s − 18-s − 19-s + 31-s + 35-s + 38-s − 40-s − 41-s − 45-s + 2·47-s − 56-s − 59-s − 62-s − 63-s + 64-s + 2·67-s − 70-s − 71-s + 72-s + 80-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(31\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  $\chi_{31} (30, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 31,\ (\ :0),\ 1)\)
\(L(\frac{1}{2})\)  \(\approx\)  \(0.2177494736\)
\(L(\frac12)\)  \(\approx\)  \(0.2177494736\)
\(L(1)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \neq 31$,\(F_p(T)\) is a polynomial of degree 2. If $p = 31$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad31 \( 1 - T \)
good2 \( 1 + T + T^{2} \)
3 \( ( 1 - T )( 1 + T ) \)
5 \( 1 + T + T^{2} \)
7 \( 1 + T + T^{2} \)
11 \( ( 1 - T )( 1 + T ) \)
13 \( ( 1 - T )( 1 + T ) \)
17 \( ( 1 - T )( 1 + T ) \)
19 \( 1 + T + T^{2} \)
23 \( ( 1 - T )( 1 + T ) \)
29 \( ( 1 - T )( 1 + T ) \)
37 \( ( 1 - T )( 1 + T ) \)
41 \( 1 + T + T^{2} \)
43 \( ( 1 - T )( 1 + T ) \)
47 \( ( 1 - T )^{2} \)
53 \( ( 1 - T )( 1 + T ) \)
59 \( 1 + T + T^{2} \)
61 \( ( 1 - T )( 1 + T ) \)
67 \( ( 1 - T )^{2} \)
71 \( 1 + T + T^{2} \)
73 \( ( 1 - T )( 1 + T ) \)
79 \( ( 1 - T )( 1 + T ) \)
83 \( ( 1 - T )( 1 + T ) \)
89 \( ( 1 - T )( 1 + T ) \)
97 \( 1 + T + T^{2} \)
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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

−17.30044677342347514916214466508, −16.18729920448603777480556373415, −15.36955016827026145573435350782, −13.47643814207973877644675198519, −12.31888813921751727125231993793, −10.63965438412188612144155604195, −9.568262511391337160706292941451, −8.198165198685797618624379681819, −6.92572000323394459470037275402, −4.16621475268391289462455833400, 4.16621475268391289462455833400, 6.92572000323394459470037275402, 8.198165198685797618624379681819, 9.568262511391337160706292941451, 10.63965438412188612144155604195, 12.31888813921751727125231993793, 13.47643814207973877644675198519, 15.36955016827026145573435350782, 16.18729920448603777480556373415, 17.30044677342347514916214466508

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