↑

Properties

Label 1-1157-1157.1038-r0-0-0
Degree $1$
Conductor $1157$
Sign $0.929 + 0.369i$
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  + (−0.814 + 0.580i)2-s + (0.436 − 0.899i)3-s + (0.327 − 0.945i)4-s + (−0.959 + 0.281i)5-s + (0.165 + 0.986i)6-s + (0.999 − 0.0237i)7-s + (0.281 + 0.959i)8-s + (−0.618 − 0.786i)9-s + (0.618 − 0.786i)10-s + (0.690 + 0.723i)11-s + (−0.707 − 0.707i)12-s + (−0.800 + 0.599i)14-s + (−0.165 + 0.986i)15-s + (−0.786 − 0.618i)16-s + (0.814 + 0.580i)17-s + (0.959 + 0.281i)18-s + ⋯
L(s)  = 1  + (−0.814 + 0.580i)2-s + (0.436 − 0.899i)3-s + (0.327 − 0.945i)4-s + (−0.959 + 0.281i)5-s + (0.165 + 0.986i)6-s + (0.999 − 0.0237i)7-s + (0.281 + 0.959i)8-s + (−0.618 − 0.786i)9-s + (0.618 − 0.786i)10-s + (0.690 + 0.723i)11-s + (−0.707 − 0.707i)12-s + (−0.800 + 0.599i)14-s + (−0.165 + 0.986i)15-s + (−0.786 − 0.618i)16-s + (0.814 + 0.580i)17-s + (0.959 + 0.281i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.929 + 0.369i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.929 + 0.369i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1157\)    =    \(13 \cdot 89\)
Sign: $0.929 + 0.369i$
Analytic conductor: \(5.37308\)
Root analytic conductor: \(5.37308\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1157} (1038, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1157,\ (0:\ ),\ 0.929 + 0.369i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.134876377 + 0.2171767638i\)
\(L(\frac12)\) \(\approx\) \(1.134876377 + 0.2171767638i\)
\(L(1)\) \(\approx\) \(0.8641665938 + 0.04693246741i\)
\(L(1)\) \(\approx\) \(0.8641665938 + 0.04693246741i\)

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 \( 1 + (0.436 - 0.899i)T \)
5 \( 1 + (-0.959 + 0.281i)T \)
7 \( 1 + (0.999 - 0.0237i)T \)
11 \( 1 + (0.690 + 0.723i)T \)
17 \( 1 + (0.814 + 0.580i)T \)
19 \( 1 + (0.992 + 0.118i)T \)
23 \( 1 + (-0.393 - 0.919i)T \)
29 \( 1 + (0.520 + 0.853i)T \)
31 \( 1 + (-0.800 + 0.599i)T \)
37 \( 1 + (-0.965 + 0.258i)T \)
41 \( 1 + (0.560 - 0.828i)T \)
43 \( 1 + (-0.520 + 0.853i)T \)
47 \( 1 + (0.654 + 0.755i)T \)
53 \( 1 + (0.755 + 0.654i)T \)
59 \( 1 + (-0.899 + 0.436i)T \)
61 \( 1 + (-0.739 + 0.672i)T \)
67 \( 1 + (0.945 - 0.327i)T \)
71 \( 1 + (0.235 + 0.971i)T \)
73 \( 1 + (-0.989 - 0.142i)T \)
79 \( 1 + (-0.989 - 0.142i)T \)
83 \( 1 + (0.936 + 0.349i)T \)
97 \( 1 + (-0.690 + 0.723i)T \)
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

−21.067565756342716567275074306319, −20.32056235105738219861345228956, −19.86343371694258158408631642869, −19.06893887375623274434527211895, −18.33226447888514213779110828931, −17.24704053923059627529616446093, −16.63798323057513752819696870505, −15.91185712753884165454795711691, −15.275520946376953655136481096970, −14.24365345232393214537656390746, −13.55316399640436946287594672971, −12.088492384352745155091267150096, −11.582940984894278135077486442712, −11.09820510462711395435443029472, −10.08190630599858017163845824538, −9.25317830087660466871688092700, −8.59218557241434492885211587154, −7.868393546569731370747550635286, −7.31296200857460521828396912506, −5.60379209278982852033920072281, −4.63080407590134169944224241469, −3.70801353198671450812475837559, −3.224289815475818102208387405567, −1.91721654669231594241400754754, −0.74729184042681327406651809163, 1.04301377393377937479787542943, 1.70379887434031387117955531447, 2.91986416057276114050939554463, 4.13078339869869556142599144919, 5.2410682042644059815673198661, 6.31945450072202953638982858845, 7.28458787127378001887375486686, 7.545561753751396332454942060801, 8.451172035719961043706781872083, 8.98598928056775037130031428742, 10.22009720170270941703127220934, 11.05104508928295685637443283594, 11.99925588225471056257843230766, 12.37144705964982200897931310105, 14.01852547007827256152281901153, 14.44471880677051067527458293345, 14.97503395509689269211449197906, 15.90429628866280854801407044270, 16.858503401477433135355805059404, 17.63614609147087400213030761308, 18.31857158986195461866772423483, 18.79764829999984413562345804608, 19.82759243664914839385859607692, 20.058411731381202716705385871141, 20.93143834421076896001972819138

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