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Properties

Label 1-1157-1157.401-r1-0-0
Degree $1$
Conductor $1157$
Sign $-0.103 + 0.994i$
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  + (−0.458 − 0.888i)2-s + (0.580 + 0.814i)3-s + (−0.580 + 0.814i)4-s + (−0.989 − 0.142i)5-s + (0.458 − 0.888i)6-s + (−0.371 − 0.928i)7-s + (0.989 + 0.142i)8-s + (−0.327 + 0.945i)9-s + (0.327 + 0.945i)10-s + (0.371 − 0.928i)11-s − 12-s + (−0.654 + 0.755i)14-s + (−0.458 − 0.888i)15-s + (−0.327 − 0.945i)16-s + (0.888 + 0.458i)17-s + (0.989 − 0.142i)18-s + ⋯
L(s)  = 1  + (−0.458 − 0.888i)2-s + (0.580 + 0.814i)3-s + (−0.580 + 0.814i)4-s + (−0.989 − 0.142i)5-s + (0.458 − 0.888i)6-s + (−0.371 − 0.928i)7-s + (0.989 + 0.142i)8-s + (−0.327 + 0.945i)9-s + (0.327 + 0.945i)10-s + (0.371 − 0.928i)11-s − 12-s + (−0.654 + 0.755i)14-s + (−0.458 − 0.888i)15-s + (−0.327 − 0.945i)16-s + (0.888 + 0.458i)17-s + (0.989 − 0.142i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.103 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.103 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1157\)    =    \(13 \cdot 89\)
Sign: $-0.103 + 0.994i$
Analytic conductor: \(124.336\)
Root analytic conductor: \(124.336\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1157} (401, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1157,\ (1:\ ),\ -0.103 + 0.994i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1677810157 + 0.1860563155i\)
\(L(\frac12)\) \(\approx\) \(0.1677810157 + 0.1860563155i\)
\(L(1)\) \(\approx\) \(0.6615896149 - 0.1801140604i\)
\(L(1)\) \(\approx\) \(0.6615896149 - 0.1801140604i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
89 \( 1 \)
good2 \( 1 + (-0.458 - 0.888i)T \)
3 \( 1 + (0.580 + 0.814i)T \)
5 \( 1 + (-0.989 - 0.142i)T \)
7 \( 1 + (-0.371 - 0.928i)T \)
11 \( 1 + (0.371 - 0.928i)T \)
17 \( 1 + (0.888 + 0.458i)T \)
19 \( 1 + (-0.945 - 0.327i)T \)
23 \( 1 + (-0.981 + 0.189i)T \)
29 \( 1 + (-0.786 - 0.618i)T \)
31 \( 1 + (-0.755 - 0.654i)T \)
37 \( 1 + (0.866 + 0.5i)T \)
41 \( 1 + (0.0950 - 0.995i)T \)
43 \( 1 + (0.786 - 0.618i)T \)
47 \( 1 + (0.909 - 0.415i)T \)
53 \( 1 + (0.415 - 0.909i)T \)
59 \( 1 + (-0.814 - 0.580i)T \)
61 \( 1 + (0.723 + 0.690i)T \)
67 \( 1 + (-0.814 + 0.580i)T \)
71 \( 1 + (-0.618 - 0.786i)T \)
73 \( 1 + (0.755 + 0.654i)T \)
79 \( 1 + (-0.654 + 0.755i)T \)
83 \( 1 + (0.540 + 0.841i)T \)
97 \( 1 + (0.371 + 0.928i)T \)
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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

−20.498403929406552990977397431424, −19.764959993795564114729739518766, −19.25249420348160587718635927523, −18.34977959438406988597352444689, −18.21427063915244298532723559135, −16.97110404983577514120677744505, −16.15518914658824679922720781936, −15.39109284009637923513929949258, −14.60252791504788243655854039649, −14.39135948692083086698945419259, −12.92540878033793115799660317148, −12.44593945747504022317772882573, −11.62793661872974206851611055150, −10.38895422422963343863381593779, −9.29998396877739887780521806755, −8.86207506995865026733092283453, −7.78893726544771448276486124494, −7.49155345736780513304942415511, −6.490632904830274810688241283660, −5.821897507868485522300611596175, −4.55446805430023256820269013637, −3.59213862968898421568488388989, −2.40385332116649598311825881946, −1.352016790316077222094440411347, −0.07531044645430236381336347921, 0.79807258282922794982889294773, 2.19621607942428121310360345146, 3.36641709940711660012601815540, 3.87670153738070286005798681425, 4.32830569317475195235787561106, 5.71514192746224659455544309854, 7.269274070730339921905472438233, 7.97180284848826776085048503968, 8.62206509595196110925639278785, 9.45606394335142404152081068319, 10.29803354199933502681000939583, 10.936364386140225476333699862784, 11.57869946309558097041129291995, 12.615883881427852596950342786846, 13.44702835338799687546445034523, 14.181295763754199129632092383803, 15.086975196643826756218675393422, 16.120094435366864174301456544503, 16.69040316504209520302518942872, 17.17555942209771547266218822883, 18.677024655291751417539134185109, 19.26356327866526120846247038668, 19.71537598580291210587173753589, 20.4652862409234834413665686868, 21.00673195181506348202012468718

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