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Properties

Label 1-1157-1157.213-r0-0-0
Degree $1$
Conductor $1157$
Sign $0.875 + 0.483i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.281 + 0.959i)2-s + (−0.212 − 0.977i)3-s + (−0.841 + 0.540i)4-s + (0.654 − 0.755i)5-s + (0.877 − 0.479i)6-s + (0.0713 + 0.997i)7-s + (−0.755 − 0.654i)8-s + (−0.909 + 0.415i)9-s + (0.909 + 0.415i)10-s + (−0.755 + 0.654i)11-s + (0.707 + 0.707i)12-s + (−0.936 + 0.349i)14-s + (−0.877 − 0.479i)15-s + (0.415 − 0.909i)16-s + (0.281 − 0.959i)17-s + (−0.654 − 0.755i)18-s + ⋯
L(s)  = 1  + (0.281 + 0.959i)2-s + (−0.212 − 0.977i)3-s + (−0.841 + 0.540i)4-s + (0.654 − 0.755i)5-s + (0.877 − 0.479i)6-s + (0.0713 + 0.997i)7-s + (−0.755 − 0.654i)8-s + (−0.909 + 0.415i)9-s + (0.909 + 0.415i)10-s + (−0.755 + 0.654i)11-s + (0.707 + 0.707i)12-s + (−0.936 + 0.349i)14-s + (−0.877 − 0.479i)15-s + (0.415 − 0.909i)16-s + (0.281 − 0.959i)17-s + (−0.654 − 0.755i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.875 + 0.483i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.875 + 0.483i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1157\)    =    \(13 \cdot 89\)
Sign: $0.875 + 0.483i$
Analytic conductor: \(5.37308\)
Root analytic conductor: \(5.37308\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1157} (213, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1157,\ (0:\ ),\ 0.875 + 0.483i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.428896752 + 0.3682115926i\)
\(L(\frac12)\) \(\approx\) \(1.428896752 + 0.3682115926i\)
\(L(1)\) \(\approx\) \(1.083997753 + 0.2273733911i\)
\(L(1)\) \(\approx\) \(1.083997753 + 0.2273733911i\)

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 \( 1 + (-0.212 - 0.977i)T \)
5 \( 1 + (0.654 - 0.755i)T \)
7 \( 1 + (0.0713 + 0.997i)T \)
11 \( 1 + (-0.755 + 0.654i)T \)
17 \( 1 + (0.281 - 0.959i)T \)
19 \( 1 + (0.349 - 0.936i)T \)
23 \( 1 + (-0.349 + 0.936i)T \)
29 \( 1 + (0.0713 + 0.997i)T \)
31 \( 1 + (0.936 - 0.349i)T \)
37 \( 1 + (0.707 + 0.707i)T \)
41 \( 1 + (0.212 - 0.977i)T \)
43 \( 1 + (-0.0713 + 0.997i)T \)
47 \( 1 + (0.841 - 0.540i)T \)
53 \( 1 + (0.540 - 0.841i)T \)
59 \( 1 + (0.977 + 0.212i)T \)
61 \( 1 + (-0.800 + 0.599i)T \)
67 \( 1 + (0.540 - 0.841i)T \)
71 \( 1 + (0.654 + 0.755i)T \)
73 \( 1 + (-0.909 - 0.415i)T \)
79 \( 1 + (0.909 + 0.415i)T \)
83 \( 1 + (-0.877 + 0.479i)T \)
97 \( 1 + (0.755 + 0.654i)T \)
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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

−21.16053730920741552551081564636, −20.753197609995103536507051276032, −19.87846182886197558515866340950, −18.97858820233504264570629395354, −18.24090580681406669047281737110, −17.36458270451811991065320206151, −16.76854135067602172171266377828, −15.6955205731507868047209978931, −14.71064412354506252334800473861, −14.15348976085984443340053884677, −13.55144868009767695624349618431, −12.56481639841997331878884061065, −11.47926527801538426546280134901, −10.78003777511173779767667156956, −10.20601251323469431105928280046, −9.91899725482029538135435170, −8.69172745278781176067605739815, −7.79285504131775612136559004179, −6.193796806153046844140445489007, −5.78073696226550285302820046906, −4.648014237740428001493626094120, −3.85601550770387159044790625825, −3.13245986376832968956700805965, −2.217129788159320831753736663, −0.81249954656052355530188406609, 0.83024830096206622390686897483, 2.174783054401651960840910477746, 2.97254359347976540698423672260, 4.72874303137544538665584175142, 5.27979010528994093350981604626, 5.86160442357401076931095463550, 6.835114203799329747979004502867, 7.61417584408322455563351082944, 8.42212689585441607372795400688, 9.15636088484089669727371950584, 9.914972901727932913354779591668, 11.54594713152078883037334528450, 12.19971526773288192166334624112, 12.91447792863594242522096002674, 13.49229041982833949139927059631, 14.17585096669044750775720745572, 15.23501545699664451491151256308, 15.927287696900488700357758911406, 16.723974459773999935658611610788, 17.66828577635669221664159978169, 17.99689179688938515183970698603, 18.61819553401744064942840155188, 19.72324384518732751292848274400, 20.7039864383108397230288620625, 21.513905509032891989666865047446

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