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Properties

Label 2-1183-7.4-c1-0-75
Degree $2$
Conductor $1183$
Sign $0.171 + 0.985i$
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  + (0.425 + 0.737i)2-s + (−0.330 + 0.572i)3-s + (0.637 − 1.10i)4-s + (−1.72 − 2.98i)5-s − 0.562·6-s + (0.751 − 2.53i)7-s + 2.78·8-s + (1.28 + 2.21i)9-s + (1.46 − 2.53i)10-s + (0.448 − 0.777i)11-s + (0.421 + 0.730i)12-s + (2.18 − 0.525i)14-s + 2.27·15-s + (−0.0891 − 0.154i)16-s + (−0.968 + 1.67i)17-s + (−1.09 + 1.88i)18-s + ⋯
L(s)  = 1  + (0.300 + 0.521i)2-s + (−0.190 + 0.330i)3-s + (0.318 − 0.552i)4-s + (−0.769 − 1.33i)5-s − 0.229·6-s + (0.284 − 0.958i)7-s + 0.985·8-s + (0.427 + 0.739i)9-s + (0.463 − 0.802i)10-s + (0.135 − 0.234i)11-s + (0.121 + 0.210i)12-s + (0.585 − 0.140i)14-s + 0.587·15-s + (−0.0222 − 0.0386i)16-s + (−0.234 + 0.406i)17-s + (−0.257 + 0.445i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.171 + 0.985i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.171 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $0.171 + 0.985i$
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: \((2,\ 1183,\ (\ :1/2),\ 0.171 + 0.985i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.662562333\)
\(L(\frac12)\) \(\approx\) \(1.662562333\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-0.751 + 2.53i)T \)
13 \( 1 \)
good2 \( 1 + (-0.425 - 0.737i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (0.330 - 0.572i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (1.72 + 2.98i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.448 + 0.777i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (0.968 - 1.67i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.519 + 0.898i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.82 + 4.89i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 1.83T + 29T^{2} \)
31 \( 1 + (-4.56 + 7.91i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-5.30 - 9.17i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 5.33T + 41T^{2} \)
43 \( 1 + 3.91T + 43T^{2} \)
47 \( 1 + (3.59 + 6.22i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4.69 + 8.12i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.255 + 0.442i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.718 + 1.24i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.22 + 7.31i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 3.44T + 71T^{2} \)
73 \( 1 + (5.45 - 9.44i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-6.04 - 10.4i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 1.51T + 83T^{2} \)
89 \( 1 + (6.80 + 11.7i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 0.506T + 97T^{2} \)
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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

−9.837614046852435546610567740313, −8.383177065833420135824180801926, −8.040379034974518936405707467572, −7.08995025487829302511947303667, −6.18600552329353693021979371218, −5.04537184900000281420034907808, −4.58660093165831719864022464889, −3.94628555658779142562442379300, −1.87410683825105462642809504180, −0.67078506398929520254109184985, 1.77495561398805324496363258538, 2.85949635619482549788466658570, 3.55772973524142360448637164894, 4.49211566159913472829748950757, 5.90020062465920350091040240450, 6.79911905327152507668528398800, 7.36452165179326105575220088369, 8.086785842112765085269717349027, 9.173745207360304084799988980232, 10.18651525860020869080294454766

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