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Properties

Label 1-1183-1183.1139-r0-0-0
Degree $1$
Conductor $1183$
Sign $0.955 - 0.295i$
Analytic cond. $5.49382$
Root an. cond. $5.49382$
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  + (0.534 − 0.845i)2-s + (−0.278 + 0.960i)3-s + (−0.428 − 0.903i)4-s + (−0.160 − 0.987i)5-s + (0.663 + 0.748i)6-s + (−0.992 − 0.120i)8-s + (−0.845 − 0.534i)9-s + (−0.919 − 0.391i)10-s + (0.534 + 0.845i)11-s + (0.987 − 0.160i)12-s + (0.992 + 0.120i)15-s + (−0.632 + 0.774i)16-s + (0.799 + 0.600i)17-s + (−0.903 + 0.428i)18-s + (−0.866 + 0.5i)19-s + (−0.822 + 0.568i)20-s + ⋯
L(s)  = 1  + (0.534 − 0.845i)2-s + (−0.278 + 0.960i)3-s + (−0.428 − 0.903i)4-s + (−0.160 − 0.987i)5-s + (0.663 + 0.748i)6-s + (−0.992 − 0.120i)8-s + (−0.845 − 0.534i)9-s + (−0.919 − 0.391i)10-s + (0.534 + 0.845i)11-s + (0.987 − 0.160i)12-s + (0.992 + 0.120i)15-s + (−0.632 + 0.774i)16-s + (0.799 + 0.600i)17-s + (−0.903 + 0.428i)18-s + (−0.866 + 0.5i)19-s + (−0.822 + 0.568i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.955 - 0.295i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.955 - 0.295i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $0.955 - 0.295i$
Analytic conductor: \(5.49382\)
Root analytic conductor: \(5.49382\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1183} (1139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1183,\ (0:\ ),\ 0.955 - 0.295i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.486017913 - 0.2247800872i\)
\(L(\frac12)\) \(\approx\) \(1.486017913 - 0.2247800872i\)
\(L(1)\) \(\approx\) \(1.112648556 - 0.2875838534i\)
\(L(1)\) \(\approx\) \(1.112648556 - 0.2875838534i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
13 \( 1 \)
good2 \( 1 + (0.534 - 0.845i)T \)
3 \( 1 + (-0.278 + 0.960i)T \)
5 \( 1 + (-0.160 - 0.987i)T \)
11 \( 1 + (0.534 + 0.845i)T \)
17 \( 1 + (0.799 + 0.600i)T \)
19 \( 1 + (-0.866 + 0.5i)T \)
23 \( 1 + (0.5 + 0.866i)T \)
29 \( 1 + (0.885 - 0.464i)T \)
31 \( 1 + (-0.979 + 0.200i)T \)
37 \( 1 + (0.316 - 0.948i)T \)
41 \( 1 + (0.239 + 0.970i)T \)
43 \( 1 + (0.748 + 0.663i)T \)
47 \( 1 + (0.903 + 0.428i)T \)
53 \( 1 + (0.799 + 0.600i)T \)
59 \( 1 + (0.774 - 0.632i)T \)
61 \( 1 + (-0.799 + 0.600i)T \)
67 \( 1 + (-0.0804 - 0.996i)T \)
71 \( 1 + (-0.239 - 0.970i)T \)
73 \( 1 + (-0.534 - 0.845i)T \)
79 \( 1 + (0.428 - 0.903i)T \)
83 \( 1 + (-0.239 + 0.970i)T \)
89 \( 1 + (0.866 - 0.5i)T \)
97 \( 1 + (-0.935 + 0.354i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−21.6917109564942624073667422511, −20.56942462140600103689384676256, −19.40274907110962485050097250742, −18.75723655206818264605583755177, −18.21857688164294380251170397283, −17.29608758451889682543618199558, −16.715387560868485174352083327120, −15.86433753053260778747028847411, −14.806154244992245827615788520, −14.27213877871766708117115350319, −13.668905663977517320389523980957, −12.79328557892120215666369367496, −11.97714301846871374277035213516, −11.31476826871607860412735481157, −10.42004759521281991976558088150, −8.932949087012778789661351915, −8.3198725371455957140358280758, −7.25737325335025518691899691544, −6.88185578799464381706806945501, −6.0673509332314889758896366142, −5.37095577503702306026229591540, −4.120906539153554248852458457122, −3.10356466701866608121728244172, −2.399280061979672073317941761638, −0.67285862748575449350960020906, 0.96125953051416315659007721341, 1.989559001342211067638382074304, 3.31946355449792770860335981253, 4.14045568523879533694400865971, 4.61935771536005327909610251876, 5.558941496328742221783020195142, 6.20449556298146713139508996677, 7.7729869745865942156305621950, 8.94950333900814949436447458744, 9.38980432578986759681101196305, 10.22644974624791737349916303059, 10.9601643806252039495461147683, 11.93904874743145024364526827222, 12.37339161389694133466125118590, 13.171144932249966762815453510234, 14.32025171366823278960692650932, 14.89520085702584467717383791395, 15.6402046316740155289894066035, 16.57463182996230961419829921866, 17.26316633509081108127304641109, 18.03172604303686575663214987002, 19.43735159540317111926754078295, 19.70648994158023419709352744282, 20.651194321603171010285534449821, 21.18924427661467128695704149684

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