Properties

Label 2-1183-7.4-c1-0-23
Degree $2$
Conductor $1183$
Sign $-0.942 + 0.333i$
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  + (0.929 + 1.60i)2-s + (−1.14 + 1.98i)3-s + (−0.726 + 1.25i)4-s + (−0.0986 − 0.170i)5-s − 4.26·6-s + (2.62 + 0.317i)7-s + 1.01·8-s + (−1.13 − 1.95i)9-s + (0.183 − 0.317i)10-s + (−2.09 + 3.62i)11-s + (−1.66 − 2.88i)12-s + (1.92 + 4.52i)14-s + 0.452·15-s + (2.39 + 4.15i)16-s + (−0.420 + 0.728i)17-s + (2.10 − 3.64i)18-s + ⋯
L(s)  = 1  + (0.656 + 1.13i)2-s + (−0.662 + 1.14i)3-s + (−0.363 + 0.629i)4-s + (−0.0441 − 0.0764i)5-s − 1.74·6-s + (0.992 + 0.120i)7-s + 0.359·8-s + (−0.377 − 0.653i)9-s + (0.0579 − 0.100i)10-s + (−0.630 + 1.09i)11-s + (−0.481 − 0.833i)12-s + (0.515 + 1.20i)14-s + 0.116·15-s + (0.599 + 1.03i)16-s + (−0.102 + 0.176i)17-s + (0.495 − 0.858i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.942 + 0.333i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.942 + 0.333i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $-0.942 + 0.333i$
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: \((2,\ 1183,\ (\ :1/2),\ -0.942 + 0.333i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.944568569\)
\(L(\frac12)\) \(\approx\) \(1.944568569\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-2.62 - 0.317i)T \)
13 \( 1 \)
good2 \( 1 + (-0.929 - 1.60i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (1.14 - 1.98i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (0.0986 + 0.170i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (2.09 - 3.62i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (0.420 - 0.728i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.675 - 1.17i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.05 - 3.56i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 8.23T + 29T^{2} \)
31 \( 1 + (0.640 - 1.10i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.52 - 2.63i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 5.39T + 41T^{2} \)
43 \( 1 - 5.32T + 43T^{2} \)
47 \( 1 + (5.83 + 10.1i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.32 - 4.02i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.02 + 5.24i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.68 - 9.84i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6.69 + 11.6i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 5.97T + 71T^{2} \)
73 \( 1 + (-1.94 + 3.36i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-5.36 - 9.29i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 3.07T + 83T^{2} \)
89 \( 1 + (5.99 + 10.3i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 19.4T + 97T^{2} \)
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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

−10.27413860056184017620712825376, −9.493992073978850834225481215638, −8.332988499327013006734213854504, −7.60613540641135310328377567556, −6.81659133667013866249936409977, −5.63165633209787703157459808418, −5.16256187336013472240965886400, −4.59085409939282791594928014748, −3.78446601532801976317602027984, −1.89697077705853057170395096219, 0.75601250197995521264268409341, 1.76716955536567402027773441317, 2.78315344896630081863370679257, 3.90403206236618871985449378529, 5.08268411905209656819411760733, 5.63341753106246889909119676801, 6.84994400435902074982766628434, 7.59942012373037156124164660568, 8.303081510276013614108914737896, 9.502928969324805973755371058432

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