↑

Properties

Label 1-1157-1157.1048-r1-0-0
Degree $1$
Conductor $1157$
Sign $0.254 + 0.967i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.281 − 0.959i)2-s + (−0.540 − 0.841i)3-s + (−0.841 − 0.540i)4-s + (−0.755 + 0.654i)5-s + (−0.959 + 0.281i)6-s + (0.654 + 0.755i)7-s + (−0.755 + 0.654i)8-s + (−0.415 + 0.909i)9-s + (0.415 + 0.909i)10-s + (0.755 + 0.654i)11-s + i·12-s + (0.909 − 0.415i)14-s + (0.959 + 0.281i)15-s + (0.415 + 0.909i)16-s + (−0.959 + 0.281i)17-s + (0.755 + 0.654i)18-s + ⋯
L(s)  = 1  + (0.281 − 0.959i)2-s + (−0.540 − 0.841i)3-s + (−0.841 − 0.540i)4-s + (−0.755 + 0.654i)5-s + (−0.959 + 0.281i)6-s + (0.654 + 0.755i)7-s + (−0.755 + 0.654i)8-s + (−0.415 + 0.909i)9-s + (0.415 + 0.909i)10-s + (0.755 + 0.654i)11-s + i·12-s + (0.909 − 0.415i)14-s + (0.959 + 0.281i)15-s + (0.415 + 0.909i)16-s + (−0.959 + 0.281i)17-s + (0.755 + 0.654i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.254 + 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.254 + 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1157\)    =    \(13 \cdot 89\)
Sign: $0.254 + 0.967i$
Analytic conductor: \(124.336\)
Root analytic conductor: \(124.336\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1157} (1048, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1157,\ (1:\ ),\ 0.254 + 0.967i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6316535088 + 0.4868055008i\)
\(L(\frac12)\) \(\approx\) \(0.6316535088 + 0.4868055008i\)
\(L(1)\) \(\approx\) \(0.7636686123 - 0.2828164607i\)
\(L(1)\) \(\approx\) \(0.7636686123 - 0.2828164607i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
89 \( 1 \)
good2 \( 1 + (0.281 - 0.959i)T \)
3 \( 1 + (-0.540 - 0.841i)T \)
5 \( 1 + (-0.755 + 0.654i)T \)
7 \( 1 + (0.654 + 0.755i)T \)
11 \( 1 + (0.755 + 0.654i)T \)
17 \( 1 + (-0.959 + 0.281i)T \)
19 \( 1 + (-0.415 + 0.909i)T \)
23 \( 1 + (0.909 + 0.415i)T \)
29 \( 1 + (0.755 - 0.654i)T \)
31 \( 1 + (-0.415 - 0.909i)T \)
37 \( 1 + T \)
41 \( 1 + (0.841 + 0.540i)T \)
43 \( 1 + (-0.755 - 0.654i)T \)
47 \( 1 + (-0.540 + 0.841i)T \)
53 \( 1 + (-0.841 + 0.540i)T \)
59 \( 1 + (0.841 + 0.540i)T \)
61 \( 1 + (-0.989 + 0.142i)T \)
67 \( 1 + (0.540 + 0.841i)T \)
71 \( 1 + (0.755 + 0.654i)T \)
73 \( 1 + (0.909 - 0.415i)T \)
79 \( 1 + (-0.415 - 0.909i)T \)
83 \( 1 + (-0.959 + 0.281i)T \)
97 \( 1 + (0.755 - 0.654i)T \)
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

−21.27281062796493325537213042321, −20.16299054761735389694119471898, −19.61099485363155345407017887197, −18.24009471363346146611353982744, −17.46473192836904468246157620714, −16.864418075518154937271579867457, −16.31106539225834938727236572385, −15.62028581725049675471666090940, −14.85063053860443111670458985645, −14.18088687043249751192244673146, −13.18979744034549959975517827676, −12.35613201527413352919429238136, −11.30829453026062813579415414575, −10.95083784198640482028069535005, −9.529356988568034269346636985241, −8.78721848998474630551630384178, −8.26636290587028598706032579645, −7.00814050936138376354160556478, −6.49273979495310784646173699180, −5.13895820433486716101155137440, −4.72408949815053925239683079433, −4.03598390200285394353269797219, −3.20515178847816712820596653990, −0.96871806763027854295739776264, −0.2116813685375268386511620455, 1.12065096698508867144825328945, 2.0428624709325904733705555974, 2.75475249641233040735333093145, 4.0709368266112966589959769295, 4.73180194981449316010796986613, 5.887181339586131207129072945994, 6.56615624361387323029637821363, 7.71299768051391539173063883216, 8.42383945122489615629962360559, 9.43263579986976621653351219693, 10.57720945935360840690767952747, 11.34469668083100182947278323928, 11.68050182842169036903237601526, 12.46814039639770089629965574428, 13.12145566442736731885493345859, 14.25661856078096870500240613979, 14.79162786554957692659115365993, 15.54665241297787585595995421422, 16.97563832051570229654604982580, 17.70558525865006924097939244458, 18.34670690743796033681610832685, 18.988697575593229481353623176512, 19.577287509263528772598100673, 20.30891355788927226904149011126, 21.39680847698338332675003642760

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