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Properties

Label 2-1183-91.44-c0-0-0
Degree $2$
Conductor $1183$
Sign $0.999 - 0.0311i$
Analytic cond. $0.590393$
Root an. cond. $0.768370$
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  + (0.258 + 0.965i)2-s + (−0.5 − 0.866i)3-s + (−0.258 − 0.965i)5-s + (0.707 − 0.707i)6-s + (0.707 + 0.707i)7-s + (0.707 + 0.707i)8-s + (0.866 − 0.499i)10-s + (−0.258 + 0.965i)11-s + (−0.500 + 0.866i)14-s + (−0.707 + 0.707i)15-s + (−0.5 + 0.866i)16-s + (0.866 − 0.5i)17-s + (−0.258 − 0.965i)19-s + (0.258 − 0.965i)21-s − 22-s + (−0.866 − 0.5i)23-s + ⋯
L(s)  = 1  + (0.258 + 0.965i)2-s + (−0.5 − 0.866i)3-s + (−0.258 − 0.965i)5-s + (0.707 − 0.707i)6-s + (0.707 + 0.707i)7-s + (0.707 + 0.707i)8-s + (0.866 − 0.499i)10-s + (−0.258 + 0.965i)11-s + (−0.500 + 0.866i)14-s + (−0.707 + 0.707i)15-s + (−0.5 + 0.866i)16-s + (0.866 − 0.5i)17-s + (−0.258 − 0.965i)19-s + (0.258 − 0.965i)21-s − 22-s + (−0.866 − 0.5i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0311i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0311i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $0.999 - 0.0311i$
Analytic conductor: \(0.590393\)
Root analytic conductor: \(0.768370\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1183} (408, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1183,\ (\ :0),\ 0.999 - 0.0311i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.216085821\)
\(L(\frac12)\) \(\approx\) \(1.216085821\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-0.707 - 0.707i)T \)
13 \( 1 \)
good2 \( 1 + (-0.258 - 0.965i)T + (-0.866 + 0.5i)T^{2} \)
3 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
5 \( 1 + (0.258 + 0.965i)T + (-0.866 + 0.5i)T^{2} \)
11 \( 1 + (0.258 - 0.965i)T + (-0.866 - 0.5i)T^{2} \)
17 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (0.258 + 0.965i)T + (-0.866 + 0.5i)T^{2} \)
23 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + (-0.965 - 0.258i)T + (0.866 + 0.5i)T^{2} \)
37 \( 1 + (-0.965 + 0.258i)T + (0.866 - 0.5i)T^{2} \)
41 \( 1 - iT^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (0.965 - 0.258i)T + (0.866 - 0.5i)T^{2} \)
53 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
59 \( 1 + (0.258 - 0.965i)T + (-0.866 - 0.5i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.965 + 0.258i)T + (0.866 + 0.5i)T^{2} \)
71 \( 1 - iT^{2} \)
73 \( 1 + (-0.258 + 0.965i)T + (-0.866 - 0.5i)T^{2} \)
79 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 + (0.965 - 0.258i)T + (0.866 - 0.5i)T^{2} \)
97 \( 1 + (1.41 - 1.41i)T - iT^{2} \)
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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

−9.831485793019243294592979057017, −8.838866405799324888291807316430, −7.972981128929056056520981223293, −7.51237800041400051968893644272, −6.58144666479003366134719317804, −5.87998137731799560674600400937, −4.95441740777550764273947237966, −4.52374474167369936385697124555, −2.40607410544921922877263731024, −1.30366030796550385958362119253, 1.57031237878080652670915735767, 2.99235024046009768568421986929, 3.76761433296570823770375489454, 4.42292790004324743908494914586, 5.57126986835363221846225213840, 6.53472137231597079122865854482, 7.71741580911029635635454859667, 8.070289125157541572281835619275, 9.825705797871159970777218029102, 10.20289736878152789957027105800

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