↑

Properties

Label 1-1157-1157.1022-r0-0-0
Degree $1$
Conductor $1157$
Sign $0.979 - 0.203i$
Analytic cond. $5.37308$
Root an. cond. $5.37308$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.989 − 0.142i)2-s + (−0.479 + 0.877i)3-s + (0.959 + 0.281i)4-s + (−0.415 + 0.909i)5-s + (0.599 − 0.800i)6-s + (−0.936 + 0.349i)7-s + (−0.909 − 0.415i)8-s + (−0.540 − 0.841i)9-s + (0.540 − 0.841i)10-s + (−0.909 + 0.415i)11-s + (−0.707 + 0.707i)12-s + (0.977 − 0.212i)14-s + (−0.599 − 0.800i)15-s + (0.841 + 0.540i)16-s + (−0.989 + 0.142i)17-s + (0.415 + 0.909i)18-s + ⋯
L(s)  = 1  + (−0.989 − 0.142i)2-s + (−0.479 + 0.877i)3-s + (0.959 + 0.281i)4-s + (−0.415 + 0.909i)5-s + (0.599 − 0.800i)6-s + (−0.936 + 0.349i)7-s + (−0.909 − 0.415i)8-s + (−0.540 − 0.841i)9-s + (0.540 − 0.841i)10-s + (−0.909 + 0.415i)11-s + (−0.707 + 0.707i)12-s + (0.977 − 0.212i)14-s + (−0.599 − 0.800i)15-s + (0.841 + 0.540i)16-s + (−0.989 + 0.142i)17-s + (0.415 + 0.909i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.979 - 0.203i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.979 - 0.203i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1157\)    =    \(13 \cdot 89\)
Sign: $0.979 - 0.203i$
Analytic conductor: \(5.37308\)
Root analytic conductor: \(5.37308\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1157} (1022, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1157,\ (0:\ ),\ 0.979 - 0.203i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1354467063 + 0.01395201167i\)
\(L(\frac12)\) \(\approx\) \(0.1354467063 + 0.01395201167i\)
\(L(1)\) \(\approx\) \(0.3293793668 + 0.1589683972i\)
\(L(1)\) \(\approx\) \(0.3293793668 + 0.1589683972i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
89 \( 1 \)
good2 \( 1 + (-0.989 - 0.142i)T \)
3 \( 1 + (-0.479 + 0.877i)T \)
5 \( 1 + (-0.415 + 0.909i)T \)
7 \( 1 + (-0.936 + 0.349i)T \)
11 \( 1 + (-0.909 + 0.415i)T \)
17 \( 1 + (-0.989 + 0.142i)T \)
19 \( 1 + (0.212 - 0.977i)T \)
23 \( 1 + (-0.212 + 0.977i)T \)
29 \( 1 + (-0.936 + 0.349i)T \)
31 \( 1 + (-0.977 + 0.212i)T \)
37 \( 1 + (-0.707 + 0.707i)T \)
41 \( 1 + (0.479 + 0.877i)T \)
43 \( 1 + (0.936 + 0.349i)T \)
47 \( 1 + (-0.959 - 0.281i)T \)
53 \( 1 + (-0.281 - 0.959i)T \)
59 \( 1 + (0.877 - 0.479i)T \)
61 \( 1 + (-0.0713 - 0.997i)T \)
67 \( 1 + (-0.281 - 0.959i)T \)
71 \( 1 + (-0.415 - 0.909i)T \)
73 \( 1 + (-0.540 + 0.841i)T \)
79 \( 1 + (0.540 - 0.841i)T \)
83 \( 1 + (-0.599 + 0.800i)T \)
97 \( 1 + (0.909 + 0.415i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−20.845052607539113050378060369625, −20.37400732653656387771257015452, −19.49090253817369575245612641701, −19.00238772755176989233244675135, −18.26386389687676795029812648993, −17.4687028382511752091691058931, −16.59687180356248938214452438891, −16.252350617233978654530328550707, −15.57408758355626241541861662665, −14.251023113297655703611306592857, −13.142454139278309377613105386238, −12.68501882466540632126799773841, −11.8679166355880780092179375758, −10.957746440508962572809374691794, −10.315390302669948775215957986816, −9.17115799038684159427569422556, −8.514419149030821287554048958700, −7.613344865833467740519219402327, −7.11533526744222814827736175589, −5.97551778365618021248845547708, −5.50724718977338715540335544380, −4.05621883725020057016690021414, −2.73606236351385787607861389546, −1.77703479266305876273875645114, −0.588273739104688767880200939371, 0.141574643501927676165145205, 2.127815829541031532006392347703, 3.06562307243197116726843355321, 3.6625291907212610904620357843, 5.038027545688335554566992282889, 6.10965041518050506075404326206, 6.8116381197921921461274613401, 7.594961690875136805061561311759, 8.747778548734463847619217369310, 9.55313857383944764824264160911, 10.073160868993389347524836631102, 11.057051586134321274677370819292, 11.28792457782721009832617170730, 12.38211687931792804909601448971, 13.24696484722930772426773543668, 14.76163883033409068684720180603, 15.4185111476022268258040069311, 15.82383016089999378026171443898, 16.490377050198608848090554354617, 17.70015475770504741243568154769, 17.959266757422454481073638419332, 18.914875768913045356660551963715, 19.70680801802077687410429741699, 20.265403405251493466826437114535, 21.31047965044832962418234499442

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