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Properties

Label 1-1157-1157.93-r1-0-0
Degree $1$
Conductor $1157$
Sign $-0.998 + 0.0450i$
Analytic cond. $124.336$
Root an. cond. $124.336$
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  + (0.814 − 0.580i)2-s + (−0.327 − 0.945i)3-s + (0.327 − 0.945i)4-s + (0.281 + 0.959i)5-s + (−0.814 − 0.580i)6-s + (−0.690 + 0.723i)7-s + (−0.281 − 0.959i)8-s + (−0.786 + 0.618i)9-s + (0.786 + 0.618i)10-s + (0.690 + 0.723i)11-s − 12-s + (−0.142 + 0.989i)14-s + (0.814 − 0.580i)15-s + (−0.786 − 0.618i)16-s + (−0.580 + 0.814i)17-s + (−0.281 + 0.959i)18-s + ⋯
L(s)  = 1  + (0.814 − 0.580i)2-s + (−0.327 − 0.945i)3-s + (0.327 − 0.945i)4-s + (0.281 + 0.959i)5-s + (−0.814 − 0.580i)6-s + (−0.690 + 0.723i)7-s + (−0.281 − 0.959i)8-s + (−0.786 + 0.618i)9-s + (0.786 + 0.618i)10-s + (0.690 + 0.723i)11-s − 12-s + (−0.142 + 0.989i)14-s + (0.814 − 0.580i)15-s + (−0.786 − 0.618i)16-s + (−0.580 + 0.814i)17-s + (−0.281 + 0.959i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.998 + 0.0450i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.998 + 0.0450i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1157\)    =    \(13 \cdot 89\)
Sign: $-0.998 + 0.0450i$
Analytic conductor: \(124.336\)
Root analytic conductor: \(124.336\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1157} (93, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1157,\ (1:\ ),\ -0.998 + 0.0450i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.02451414815 - 1.088870412i\)
\(L(\frac12)\) \(\approx\) \(0.02451414815 - 1.088870412i\)
\(L(1)\) \(\approx\) \(1.110325555 - 0.5406396766i\)
\(L(1)\) \(\approx\) \(1.110325555 - 0.5406396766i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
89 \( 1 \)
good2 \( 1 + (0.814 - 0.580i)T \)
3 \( 1 + (-0.327 - 0.945i)T \)
5 \( 1 + (0.281 + 0.959i)T \)
7 \( 1 + (-0.690 + 0.723i)T \)
11 \( 1 + (0.690 + 0.723i)T \)
17 \( 1 + (-0.580 + 0.814i)T \)
19 \( 1 + (0.618 + 0.786i)T \)
23 \( 1 + (-0.928 - 0.371i)T \)
29 \( 1 + (0.235 - 0.971i)T \)
31 \( 1 + (0.989 + 0.142i)T \)
37 \( 1 + (-0.866 - 0.5i)T \)
41 \( 1 + (-0.189 - 0.981i)T \)
43 \( 1 + (-0.235 - 0.971i)T \)
47 \( 1 + (0.755 - 0.654i)T \)
53 \( 1 + (-0.654 + 0.755i)T \)
59 \( 1 + (-0.945 - 0.327i)T \)
61 \( 1 + (0.0475 - 0.998i)T \)
67 \( 1 + (-0.945 + 0.327i)T \)
71 \( 1 + (0.971 - 0.235i)T \)
73 \( 1 + (-0.989 - 0.142i)T \)
79 \( 1 + (-0.142 + 0.989i)T \)
83 \( 1 + (0.909 - 0.415i)T \)
97 \( 1 + (0.690 - 0.723i)T \)
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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

−21.68608353883929613702628288039, −20.84570363997562285976209968431, −20.13247321516754057508230596006, −19.67702816180070661939551287427, −17.90367481480085152592884545324, −17.27771876490534930235626607974, −16.5658749494841355345132771050, −16.04593809706796667677028616417, −15.60897866706548693188386046871, −14.34284217131433571339561439671, −13.73629896293298037563078490772, −13.136353820429925575377451441075, −11.98310999849182910241403541569, −11.53485261644739942654924672020, −10.43077446378258711000092895081, −9.384695652936593807345115026377, −8.89550968206463104454542248497, −7.8229468834056089066581812909, −6.603883310743276083182258475566, −6.0767034276098460461025538075, −5.016223639103340404039840034598, −4.51745623019218487443241838574, −3.61125361338424333201514622112, −2.849903064228557434400728636703, −1.039963509966801182000113287829, 0.17355752614848582090797960796, 1.74287564474860908378798412051, 2.192260465947029648914502460603, 3.17257288902629532305910205290, 4.10875140515604701214737080828, 5.45893966039796093397739163177, 6.1955340751617524627457817631, 6.5845312645894549606449032231, 7.51373311617074157336663585402, 8.82775335166904520029540610321, 9.934865283439187221671010571159, 10.49062949301006809892363543746, 11.56776857084991559207539215243, 12.170609740158618141702869749388, 12.63491677480392400151060246765, 13.81705405007783841380985551025, 14.049339218120242548005287785248, 15.15496162834335692144861006582, 15.69598434490580600033655124194, 17.024603615438446767517884260554, 17.84942524352298042423755924591, 18.63102272888048396613290331368, 19.15099010522406310334911206230, 19.77534033661471591781248726232, 20.68359795337699513886638860887

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