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Properties

Label 1-1157-1157.180-r1-0-0
Degree $1$
Conductor $1157$
Sign $-0.852 + 0.522i$
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  + (−0.998 + 0.0475i)2-s + (−0.995 − 0.0950i)3-s + (0.995 − 0.0950i)4-s + (0.989 − 0.142i)5-s + (0.998 + 0.0475i)6-s + (0.618 + 0.786i)7-s + (−0.989 + 0.142i)8-s + (0.981 + 0.189i)9-s + (−0.981 + 0.189i)10-s + (−0.618 + 0.786i)11-s − 12-s + (−0.654 − 0.755i)14-s + (−0.998 + 0.0475i)15-s + (0.981 − 0.189i)16-s + (−0.0475 + 0.998i)17-s + (−0.989 − 0.142i)18-s + ⋯
L(s)  = 1  + (−0.998 + 0.0475i)2-s + (−0.995 − 0.0950i)3-s + (0.995 − 0.0950i)4-s + (0.989 − 0.142i)5-s + (0.998 + 0.0475i)6-s + (0.618 + 0.786i)7-s + (−0.989 + 0.142i)8-s + (0.981 + 0.189i)9-s + (−0.981 + 0.189i)10-s + (−0.618 + 0.786i)11-s − 12-s + (−0.654 − 0.755i)14-s + (−0.998 + 0.0475i)15-s + (0.981 − 0.189i)16-s + (−0.0475 + 0.998i)17-s + (−0.989 − 0.142i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.852 + 0.522i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.852 + 0.522i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1157\)    =    \(13 \cdot 89\)
Sign: $-0.852 + 0.522i$
Analytic conductor: \(124.336\)
Root analytic conductor: \(124.336\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1157} (180, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1157,\ (1:\ ),\ -0.852 + 0.522i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2694945387 + 0.9559240124i\)
\(L(\frac12)\) \(\approx\) \(0.2694945387 + 0.9559240124i\)
\(L(1)\) \(\approx\) \(0.6227004023 + 0.2236399095i\)
\(L(1)\) \(\approx\) \(0.6227004023 + 0.2236399095i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
89 \( 1 \)
good2 \( 1 + (-0.998 + 0.0475i)T \)
3 \( 1 + (-0.995 - 0.0950i)T \)
5 \( 1 + (0.989 - 0.142i)T \)
7 \( 1 + (0.618 + 0.786i)T \)
11 \( 1 + (-0.618 + 0.786i)T \)
17 \( 1 + (-0.0475 + 0.998i)T \)
19 \( 1 + (-0.189 + 0.981i)T \)
23 \( 1 + (0.327 + 0.945i)T \)
29 \( 1 + (0.928 + 0.371i)T \)
31 \( 1 + (0.755 - 0.654i)T \)
37 \( 1 + (0.866 + 0.5i)T \)
41 \( 1 + (-0.814 + 0.580i)T \)
43 \( 1 + (-0.928 + 0.371i)T \)
47 \( 1 + (-0.909 - 0.415i)T \)
53 \( 1 + (0.415 + 0.909i)T \)
59 \( 1 + (0.0950 + 0.995i)T \)
61 \( 1 + (0.235 + 0.971i)T \)
67 \( 1 + (0.0950 - 0.995i)T \)
71 \( 1 + (0.371 + 0.928i)T \)
73 \( 1 + (-0.755 + 0.654i)T \)
79 \( 1 + (-0.654 - 0.755i)T \)
83 \( 1 + (-0.540 + 0.841i)T \)
97 \( 1 + (-0.618 - 0.786i)T \)
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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

−20.93515428498186150332696499703, −20.02239659895218454330495398383, −18.88788595372149437562907707999, −18.248199864966288664775931223585, −17.639096731132142323240738719582, −17.137586482427466766232820495852, −16.359166954167690782164445280838, −15.76449937773403072168854322932, −14.62840860935008870974241536948, −13.64837184802303432733617799426, −12.893000660485652024541112485000, −11.70715562896874690337405839768, −11.107750727053581632688665770328, −10.42912877119737289538275807284, −9.931181654188163071258017333422, −8.892350110685985188938266623110, −7.98681156584689326451579926503, −6.851101134623411989346623976722, −6.53704019321856144235364824945, −5.36171014731046756001740248550, −4.70901573551928627556752772681, −3.11595992593391573550308912427, −2.11850585012485851927815828828, −0.95919236605022581521804659627, −0.37692791266367690052494637653, 1.29962662878849659679595732079, 1.74149369437004790916042645047, 2.77061656908034406829260932636, 4.546322595367873395015750704979, 5.49699206807714425283229339403, 6.022977500925486813996470753256, 6.85876379001519593059118998821, 7.92005286032969984515755576658, 8.61222946093503584943767864424, 9.89321328308379665445774697485, 10.0524851114414028435665551253, 11.04748095681319562677830189527, 11.86970624359161243911710308827, 12.53584713217287678964978646815, 13.37215697089978227200570028352, 14.81942164017422872765351851976, 15.27520666295297129299402828221, 16.28824116108742570474615663104, 17.06840561559899249692743219497, 17.543236394997615816720910263694, 18.27522503218240667191826032038, 18.60446025882137539954083763948, 19.73885329115902863971067779262, 20.805339740109437499829689538603, 21.389236979148354170393215117673

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