↑

Properties

Label 1-1157-1157.70-r0-0-0
Degree $1$
Conductor $1157$
Sign $0.861 - 0.507i$
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  + (0.755 + 0.654i)2-s + (0.800 − 0.599i)3-s + (0.142 + 0.989i)4-s + (−0.841 − 0.540i)5-s + (0.997 + 0.0713i)6-s + (0.977 − 0.212i)7-s + (−0.540 + 0.841i)8-s + (0.281 − 0.959i)9-s + (−0.281 − 0.959i)10-s + (−0.540 − 0.841i)11-s + (0.707 + 0.707i)12-s + (0.877 + 0.479i)14-s + (−0.997 + 0.0713i)15-s + (−0.959 + 0.281i)16-s + (0.755 − 0.654i)17-s + (0.841 − 0.540i)18-s + ⋯
L(s)  = 1  + (0.755 + 0.654i)2-s + (0.800 − 0.599i)3-s + (0.142 + 0.989i)4-s + (−0.841 − 0.540i)5-s + (0.997 + 0.0713i)6-s + (0.977 − 0.212i)7-s + (−0.540 + 0.841i)8-s + (0.281 − 0.959i)9-s + (−0.281 − 0.959i)10-s + (−0.540 − 0.841i)11-s + (0.707 + 0.707i)12-s + (0.877 + 0.479i)14-s + (−0.997 + 0.0713i)15-s + (−0.959 + 0.281i)16-s + (0.755 − 0.654i)17-s + (0.841 − 0.540i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.861 - 0.507i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.861 - 0.507i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1157\)    =    \(13 \cdot 89\)
Sign: $0.861 - 0.507i$
Analytic conductor: \(5.37308\)
Root analytic conductor: \(5.37308\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1157} (70, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1157,\ (0:\ ),\ 0.861 - 0.507i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.707080873 - 0.7372648165i\)
\(L(\frac12)\) \(\approx\) \(2.707080873 - 0.7372648165i\)
\(L(1)\) \(\approx\) \(1.886798257 + 0.03037235344i\)
\(L(1)\) \(\approx\) \(1.886798257 + 0.03037235344i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
89 \( 1 \)
good2 \( 1 + (0.755 + 0.654i)T \)
3 \( 1 + (0.800 - 0.599i)T \)
5 \( 1 + (-0.841 - 0.540i)T \)
7 \( 1 + (0.977 - 0.212i)T \)
11 \( 1 + (-0.540 - 0.841i)T \)
17 \( 1 + (0.755 - 0.654i)T \)
19 \( 1 + (0.479 + 0.877i)T \)
23 \( 1 + (-0.479 - 0.877i)T \)
29 \( 1 + (0.977 - 0.212i)T \)
31 \( 1 + (-0.877 - 0.479i)T \)
37 \( 1 + (0.707 + 0.707i)T \)
41 \( 1 + (-0.800 - 0.599i)T \)
43 \( 1 + (-0.977 - 0.212i)T \)
47 \( 1 + (-0.142 - 0.989i)T \)
53 \( 1 + (0.989 + 0.142i)T \)
59 \( 1 + (0.599 - 0.800i)T \)
61 \( 1 + (0.936 - 0.349i)T \)
67 \( 1 + (0.989 + 0.142i)T \)
71 \( 1 + (-0.841 + 0.540i)T \)
73 \( 1 + (0.281 + 0.959i)T \)
79 \( 1 + (-0.281 - 0.959i)T \)
83 \( 1 + (-0.997 - 0.0713i)T \)
97 \( 1 + (0.540 - 0.841i)T \)
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

−21.48252916075156047511078136731, −20.56315874779527258413976151446, −19.88859319263079891782564681492, −19.45932801235505163939317443090, −18.41746190951239628796420530765, −17.850575455322612827357146622914, −16.2754230324168235029582673805, −15.531078810218729548232570533086, −14.93872999790822160399755949031, −14.51063025740393226188609697713, −13.67773880780927995620413490627, −12.72671253749443573711466618941, −11.834648678049955418741413653915, −11.145178503585068125780896936356, −10.392060231693126526172644276943, −9.71923798291469462324569258166, −8.594923347407927385474173135723, −7.748546218998599777043706408451, −7.01278513555727524789631456336, −5.50704010480591059831466254922, −4.78222390690500132941767334264, −4.06190159059588686286310210698, −3.20937328150988870059384610364, −2.43197596416054231281377558172, −1.46433709022500198229913158571, 0.82815794745547389036744150483, 2.17048031203548269820514532641, 3.27566678887496625141910130566, 3.91989635681375340048268204115, 4.92288184115099716613708603905, 5.7045181653949273237811727451, 6.93235347060559310611279233459, 7.65363843787075984319613630764, 8.33211925414298147984156560113, 8.57045445155593338195457219666, 10.07063215830426207903034487922, 11.51726812034161986321496274442, 11.88246543841849847216427578014, 12.7745514142040862268844963492, 13.551431628115430823278419108894, 14.25717407192858167610773584921, 14.7948536085993073528951040281, 15.70872157980980237223579299685, 16.39144394527580285796293814210, 17.13808277620036893374550879339, 18.398798214407989616583923508660, 18.623083006776960077456443093500, 20.02479868025362261050811349301, 20.5019123029242877397601182867, 21.0471653847706110246237413522

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