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Properties

Label 1-1183-1183.34-r0-0-0
Degree $1$
Conductor $1183$
Sign $0.931 - 0.363i$
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  + (0.935 − 0.354i)2-s + (−0.568 + 0.822i)3-s + (0.748 − 0.663i)4-s + (0.992 − 0.120i)5-s + (−0.239 + 0.970i)6-s + (0.464 − 0.885i)8-s + (−0.354 − 0.935i)9-s + (0.885 − 0.464i)10-s + (0.935 + 0.354i)11-s + (0.120 + 0.992i)12-s + (−0.464 + 0.885i)15-s + (0.120 − 0.992i)16-s + (0.885 + 0.464i)17-s + (−0.663 − 0.748i)18-s − i·19-s + (0.663 − 0.748i)20-s + ⋯
L(s)  = 1  + (0.935 − 0.354i)2-s + (−0.568 + 0.822i)3-s + (0.748 − 0.663i)4-s + (0.992 − 0.120i)5-s + (−0.239 + 0.970i)6-s + (0.464 − 0.885i)8-s + (−0.354 − 0.935i)9-s + (0.885 − 0.464i)10-s + (0.935 + 0.354i)11-s + (0.120 + 0.992i)12-s + (−0.464 + 0.885i)15-s + (0.120 − 0.992i)16-s + (0.885 + 0.464i)17-s + (−0.663 − 0.748i)18-s − i·19-s + (0.663 − 0.748i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.931 - 0.363i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.931 - 0.363i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $0.931 - 0.363i$
Analytic conductor: \(5.49382\)
Root analytic conductor: \(5.49382\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1183} (34, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1183,\ (0:\ ),\ 0.931 - 0.363i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.952134924 - 0.5551912709i\)
\(L(\frac12)\) \(\approx\) \(2.952134924 - 0.5551912709i\)
\(L(1)\) \(\approx\) \(1.928522085 - 0.1825661645i\)
\(L(1)\) \(\approx\) \(1.928522085 - 0.1825661645i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
13 \( 1 \)
good2 \( 1 + (0.935 - 0.354i)T \)
3 \( 1 + (-0.568 + 0.822i)T \)
5 \( 1 + (0.992 - 0.120i)T \)
11 \( 1 + (0.935 + 0.354i)T \)
17 \( 1 + (0.885 + 0.464i)T \)
19 \( 1 - iT \)
23 \( 1 - T \)
29 \( 1 + (-0.354 - 0.935i)T \)
31 \( 1 + (-0.239 + 0.970i)T \)
37 \( 1 + (-0.239 + 0.970i)T \)
41 \( 1 + (-0.822 - 0.568i)T \)
43 \( 1 + (0.970 - 0.239i)T \)
47 \( 1 + (0.663 - 0.748i)T \)
53 \( 1 + (0.885 + 0.464i)T \)
59 \( 1 + (-0.992 + 0.120i)T \)
61 \( 1 + (-0.885 + 0.464i)T \)
67 \( 1 + (0.663 - 0.748i)T \)
71 \( 1 + (0.822 + 0.568i)T \)
73 \( 1 + (-0.935 - 0.354i)T \)
79 \( 1 + (-0.748 - 0.663i)T \)
83 \( 1 + (0.822 - 0.568i)T \)
89 \( 1 + iT \)
97 \( 1 + (-0.992 - 0.120i)T \)
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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

−21.55688424069054941586580926155, −20.657288907164098570382216414163, −19.86303392523664667776824123924, −18.7935223597103190507811476397, −18.12266999690478287520859556236, −17.19189084501769954449386497772, −16.70834897053530212825443546669, −16.07742758869221399548192003299, −14.62909004138087188284124511553, −14.17860822110600683295993873426, −13.608758409201022952110956564202, −12.64776098384588647960610676249, −12.15983820057459993022714499181, −11.30400494052255615044744723875, −10.485208930301809606686385170651, −9.38877736795476578739597908691, −8.20931045701405882626403425454, −7.37846311811693372385639913838, −6.57542444406073623937889536372, −5.80105449285314289910598530604, −5.48923116385987256390081713395, −4.21389326713807328209593984507, −3.139241219390292162986551604502, −2.069743110752169843749029119811, −1.3209241198309009826141615614, 1.07328862316484126810736965660, 2.09117287607368547424001517821, 3.23356079559782340862953372076, 4.10303911026684042518288795317, 4.89082670103495631979982164836, 5.70949161363472624341611140185, 6.278173418690490391715458416773, 7.14575470517138948393264355832, 8.7971472993426858054879259238, 9.64046199913885238693522189258, 10.22263717824710937131913204499, 10.92937175003988074489049768773, 11.99341011263686310656927145735, 12.32311287210557524762880518137, 13.53713279240796333966965595572, 14.11773285338685271394113897579, 14.94438357311992056560941308480, 15.567358085073489229466981388522, 16.60716825728507940843964556366, 17.10891868016866888295012600092, 17.94171795624122364061188015659, 19.01843879846140695261694912261, 20.08456500501387390448282805339, 20.537724111528159192105185064063, 21.53432529908390624859274617812

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