Properties

Label 2-1183-7.4-c1-0-3
Degree $2$
Conductor $1183$
Sign $-0.638 - 0.769i$
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  + (−0.787 − 1.36i)2-s + (−1.10 + 1.90i)3-s + (−0.239 + 0.414i)4-s + (1.88 + 3.25i)5-s + 3.46·6-s + (−0.609 − 2.57i)7-s − 2.39·8-s + (−0.924 − 1.60i)9-s + (2.96 − 5.12i)10-s + (−1.19 + 2.07i)11-s + (−0.527 − 0.913i)12-s + (−3.03 + 2.85i)14-s − 8.28·15-s + (2.36 + 4.09i)16-s + (0.152 − 0.264i)17-s + (−1.45 + 2.52i)18-s + ⋯
L(s)  = 1  + (−0.556 − 0.964i)2-s + (−0.635 + 1.10i)3-s + (−0.119 + 0.207i)4-s + (0.840 + 1.45i)5-s + 1.41·6-s + (−0.230 − 0.973i)7-s − 0.846·8-s + (−0.308 − 0.533i)9-s + (0.936 − 1.62i)10-s + (−0.361 + 0.625i)11-s + (−0.152 − 0.263i)12-s + (−0.809 + 0.763i)14-s − 2.13·15-s + (0.591 + 1.02i)16-s + (0.0369 − 0.0640i)17-s + (−0.343 + 0.594i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.638 - 0.769i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.638 - 0.769i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $-0.638 - 0.769i$
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: \((2,\ 1183,\ (\ :1/2),\ -0.638 - 0.769i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5331289803\)
\(L(\frac12)\) \(\approx\) \(0.5331289803\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (0.609 + 2.57i)T \)
13 \( 1 \)
good2 \( 1 + (0.787 + 1.36i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (1.10 - 1.90i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-1.88 - 3.25i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (1.19 - 2.07i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-0.152 + 0.264i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.23 + 2.14i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-4.16 - 7.21i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 4.97T + 29T^{2} \)
31 \( 1 + (1.82 - 3.16i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.82 + 4.89i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 3.27T + 41T^{2} \)
43 \( 1 + 9.84T + 43T^{2} \)
47 \( 1 + (6.29 + 10.9i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.92 - 6.80i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (6.69 - 11.5i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.407 + 0.705i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.05 - 1.82i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 4.60T + 71T^{2} \)
73 \( 1 + (-2.60 + 4.51i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.851 + 1.47i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 7.65T + 83T^{2} \)
89 \( 1 + (6.05 + 10.4i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 3.06T + 97T^{2} \)
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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

−10.17294866636897973251597044595, −9.800140330250422385055975144198, −9.020051223888615442879591471800, −7.41333439837204580440130136390, −6.74206136872651316409720013538, −5.81661981466989348770580436889, −4.88501517978531932986206757556, −3.61843784626847740470749065308, −2.89840959415570050192283183834, −1.66984851713494136682942777136, 0.29251496890539442228743760762, 1.57399633568615654845165315123, 2.85555286849203714545076632907, 4.86448716860190215547332869191, 5.60544666848947323209442100939, 6.27619359372281547598187104298, 6.70334681825650962884292550376, 8.175549619671415644324985687332, 8.295407211614532055050639479366, 9.184493324799464684031944938700

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