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Properties

Label 2-1183-7.2-c1-0-69
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: $\chi_{1183} (170, \cdot )$
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

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

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