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Properties

Label 2-1183-7.2-c1-0-34
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: $\chi_{1183} (170, \cdot )$
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

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

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