Properties

Label 2-331-331.330-c0-0-0
Degree $2$
Conductor $331$
Sign $-1$
Analytic cond. $0.165190$
Root an. cond. $0.406436$
Motivic weight $0$
Arithmetic yes
Rational no
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

Degree: \(2\)
Conductor: \(331\)
Sign: $-1$
Analytic conductor: \(0.165190\)
Root analytic conductor: \(0.406436\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{331} (330, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 331,\ (\ :0),\ -1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6877010839\)
\(L(\frac12)\) \(\approx\) \(0.6877010839\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$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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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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