Properties

Degree 2
Conductor 331
Sign $-1$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.41i·2-s − 1.41i·3-s − 1.00·4-s − 5-s − 2.00·6-s + 1.41i·7-s − 1.00·9-s + 1.41i·10-s − 1.41i·11-s + 1.41i·12-s + 2.00·14-s + 1.41i·15-s − 0.999·16-s + 17-s + 1.41i·18-s + 19-s + ⋯
L(s)  = 1  − 1.41i·2-s − 1.41i·3-s − 1.00·4-s − 5-s − 2.00·6-s + 1.41i·7-s − 1.00·9-s + 1.41i·10-s − 1.41i·11-s + 1.41i·12-s + 2.00·14-s + 1.41i·15-s − 0.999·16-s + 17-s + 1.41i·18-s + 19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 331 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 331 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(331\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(0\)
character  :  $\chi_{331} (330, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 331,\ (\ :0),\ -1)\)
\(L(\frac{1}{2})\)  \(\approx\)  \(0.6877010839\)
\(L(\frac12)\)  \(\approx\)  \(0.6877010839\)
\(L(1)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \neq 331$,\(F_p(T)\) is a polynomial of degree 2. If $p = 331$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad331 \( 1 + T \)
good2 \( 1 + 1.41iT - T^{2} \)
3 \( 1 + 1.41iT - T^{2} \)
5 \( 1 + T + T^{2} \)
7 \( 1 - 1.41iT - T^{2} \)
11 \( 1 + 1.41iT - T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 - T + T^{2} \)
19 \( 1 - T + T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 - 1.41iT - T^{2} \)
31 \( 1 - T + T^{2} \)
37 \( 1 + 1.41iT - T^{2} \)
41 \( 1 - 1.41iT - T^{2} \)
43 \( 1 + T + T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + T + T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - 1.41iT - T^{2} \)
67 \( 1 - T + T^{2} \)
71 \( 1 - T + T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 - T + T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 - 1.41iT - T^{2} \)
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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

−11.76683468859433818946477874790, −10.91083873293821841964060739790, −9.507420669366514883360334719268, −8.492519183158027149559572020791, −7.78215009456723663792192470023, −6.51880369802040179103685985516, −5.37791322093221991313937350617, −3.48887979307660269867872198718, −2.69167545616688574399335128965, −1.19567560639297745895689434972, 3.62076338683993143375358978340, 4.44967664222773039451235180042, 5.12872730530550645533283424764, 6.69627611116808674447654270290, 7.58691757887878700850819561572, 8.103718495431212188555855598732, 9.644803382779970298084804863080, 10.05421997598420128887414426285, 11.19950722450715336506096075360, 12.10255315698881699083287672092

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