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Properties

Label 2-1183-91.2-c0-0-0
Degree $2$
Conductor $1183$
Sign $-0.934 + 0.355i$
Analytic cond. $0.590393$
Root an. cond. $0.768370$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.707 + 0.707i)2-s + (−0.5 + 0.866i)3-s + (0.258 + 0.965i)5-s + (−0.258 − 0.965i)6-s + (0.965 − 0.258i)7-s + (−0.707 − 0.707i)8-s + (−0.866 − 0.500i)10-s + (−0.965 + 0.258i)11-s + (−0.500 + 0.866i)14-s + (−0.965 − 0.258i)15-s + 1.00·16-s + i·17-s + (−0.965 − 0.258i)19-s + (−0.258 + 0.965i)21-s + (0.500 − 0.866i)22-s + i·23-s + ⋯
L(s)  = 1  + (−0.707 + 0.707i)2-s + (−0.5 + 0.866i)3-s + (0.258 + 0.965i)5-s + (−0.258 − 0.965i)6-s + (0.965 − 0.258i)7-s + (−0.707 − 0.707i)8-s + (−0.866 − 0.500i)10-s + (−0.965 + 0.258i)11-s + (−0.500 + 0.866i)14-s + (−0.965 − 0.258i)15-s + 1.00·16-s + i·17-s + (−0.965 − 0.258i)19-s + (−0.258 + 0.965i)21-s + (0.500 − 0.866i)22-s + i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.934 + 0.355i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.934 + 0.355i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $-0.934 + 0.355i$
Analytic conductor: \(0.590393\)
Root analytic conductor: \(0.768370\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1183} (1094, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1183,\ (\ :0),\ -0.934 + 0.355i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5670447845\)
\(L(\frac12)\) \(\approx\) \(0.5670447845\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-0.965 + 0.258i)T \)
13 \( 1 \)
good2 \( 1 + (0.707 - 0.707i)T - iT^{2} \)
3 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
5 \( 1 + (-0.258 - 0.965i)T + (-0.866 + 0.5i)T^{2} \)
11 \( 1 + (0.965 - 0.258i)T + (0.866 - 0.5i)T^{2} \)
17 \( 1 - iT - T^{2} \)
19 \( 1 + (0.965 + 0.258i)T + (0.866 + 0.5i)T^{2} \)
23 \( 1 - iT - T^{2} \)
29 \( 1 + (-0.5 + 0.866i)T^{2} \)
31 \( 1 + (0.965 + 0.258i)T + (0.866 + 0.5i)T^{2} \)
37 \( 1 + (-0.707 - 0.707i)T + iT^{2} \)
41 \( 1 + (0.866 + 0.5i)T^{2} \)
43 \( 1 + (0.5 + 0.866i)T^{2} \)
47 \( 1 + (-0.965 + 0.258i)T + (0.866 - 0.5i)T^{2} \)
53 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
59 \( 1 + (-0.707 - 0.707i)T + iT^{2} \)
61 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.258 + 0.965i)T + (-0.866 + 0.5i)T^{2} \)
71 \( 1 + (-0.866 + 0.5i)T^{2} \)
73 \( 1 + (0.258 - 0.965i)T + (-0.866 - 0.5i)T^{2} \)
79 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 + (0.707 + 0.707i)T + iT^{2} \)
97 \( 1 + (-0.517 - 1.93i)T + (-0.866 + 0.5i)T^{2} \)
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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

−10.41419259000582389660049532128, −9.710860363689283136569739120897, −8.688539085144824631648876511538, −7.85409185596408994260657059194, −7.33059424916202653824837848283, −6.33382676699187325475061965178, −5.48970909531037407522253251792, −4.49549357595783839119413624270, −3.52807695980367044622788415664, −2.13508693403533543131456826203, 0.67461365766788423666098494723, 1.71397197521124138643424853070, 2.59559526096685413658842600396, 4.49125122654230348294579549498, 5.40361275444997593381040700956, 5.95012404192480278870821327300, 7.20775832589212230006192252248, 8.098742488624411978265431366823, 8.789188475665179621979146006023, 9.401268142303650549976056382992

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