Properties

Label 2-1183-7.4-c1-0-80
Degree $2$
Conductor $1183$
Sign $-0.773 + 0.634i$
Analytic cond. $9.44630$
Root an. cond. $3.07348$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.134 − 0.232i)2-s + (0.571 − 0.989i)3-s + (0.964 − 1.66i)4-s + (−1.28 − 2.21i)5-s − 0.306·6-s + (2.57 − 0.594i)7-s − 1.05·8-s + (0.846 + 1.46i)9-s + (−0.343 + 0.594i)10-s + (1.97 − 3.41i)11-s + (−1.10 − 1.90i)12-s + (−0.483 − 0.518i)14-s − 2.92·15-s + (−1.78 − 3.09i)16-s + (−0.392 + 0.679i)17-s + (0.227 − 0.393i)18-s + ⋯
L(s)  = 1  + (−0.0947 − 0.164i)2-s + (0.329 − 0.571i)3-s + (0.482 − 0.834i)4-s + (−0.572 − 0.992i)5-s − 0.125·6-s + (0.974 − 0.224i)7-s − 0.372·8-s + (0.282 + 0.488i)9-s + (−0.108 + 0.188i)10-s + (0.594 − 1.03i)11-s + (−0.318 − 0.550i)12-s + (−0.129 − 0.138i)14-s − 0.756·15-s + (−0.446 − 0.773i)16-s + (−0.0952 + 0.164i)17-s + (0.0535 − 0.0926i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.773 + 0.634i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.773 + 0.634i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $-0.773 + 0.634i$
Analytic conductor: \(9.44630\)
Root analytic conductor: \(3.07348\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1183} (508, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1183,\ (\ :1/2),\ -0.773 + 0.634i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.990615318\)
\(L(\frac12)\) \(\approx\) \(1.990615318\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-2.57 + 0.594i)T \)
13 \( 1 \)
good2 \( 1 + (0.134 + 0.232i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (-0.571 + 0.989i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (1.28 + 2.21i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.97 + 3.41i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (0.392 - 0.679i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.74 + 6.49i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.97 - 6.88i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 2.35T + 29T^{2} \)
31 \( 1 + (1.27 - 2.21i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3.37 - 5.85i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 2.43T + 41T^{2} \)
43 \( 1 + 2.24T + 43T^{2} \)
47 \( 1 + (-0.658 - 1.14i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (4.63 - 8.03i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.48 - 7.76i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.72 + 8.18i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.676 - 1.17i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 12.3T + 71T^{2} \)
73 \( 1 + (-0.384 + 0.665i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (3.09 + 5.36i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 1.07T + 83T^{2} \)
89 \( 1 + (-3.83 - 6.63i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 2.37T + 97T^{2} \)
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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

−9.139037322904756850667033224108, −8.711652958540453851451835767728, −7.83990234819868824975106576529, −7.09479735897704954356980282967, −6.12747629783327801829696250608, −5.00245419552125919121548049430, −4.48224307782368555475358842857, −2.91637822734031190425090711913, −1.60696726679092842962822827447, −0.917816753814754221781323712519, 1.96762254496594564398217461968, 3.03416615607762575263360574278, 3.99859561683435485071455768344, 4.53581848894394034660794039803, 6.19833277207552869587600321852, 6.93150148192726660783685004094, 7.59789478442387188461624716887, 8.402117723710924186071451675904, 9.109069868873937678576974955557, 10.18380407389259647960422061078

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