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Properties

Label 1-1157-1157.112-r0-0-0
Degree $1$
Conductor $1157$
Sign $-0.426 + 0.904i$
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  + (−0.755 − 0.654i)2-s + (−0.599 − 0.800i)3-s + (0.142 + 0.989i)4-s + (−0.841 − 0.540i)5-s + (−0.0713 + 0.997i)6-s + (0.212 + 0.977i)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 + (0.479 − 0.877i)14-s + (0.0713 + 0.997i)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 + (−0.599 − 0.800i)3-s + (0.142 + 0.989i)4-s + (−0.841 − 0.540i)5-s + (−0.0713 + 0.997i)6-s + (0.212 + 0.977i)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 + (0.479 − 0.877i)14-s + (0.0713 + 0.997i)15-s + (−0.959 + 0.281i)16-s + (−0.755 + 0.654i)17-s + (0.841 − 0.540i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.426 + 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.426 + 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1157\)    =    \(13 \cdot 89\)
Sign: $-0.426 + 0.904i$
Analytic conductor: \(5.37308\)
Root analytic conductor: \(5.37308\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1157} (112, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1157,\ (0:\ ),\ -0.426 + 0.904i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.08022210973 + 0.1264955262i\)
\(L(\frac12)\) \(\approx\) \(0.08022210973 + 0.1264955262i\)
\(L(1)\) \(\approx\) \(0.4466447681 - 0.1278147489i\)
\(L(1)\) \(\approx\) \(0.4466447681 - 0.1278147489i\)

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 \( 1 + (-0.599 - 0.800i)T \)
5 \( 1 + (-0.841 - 0.540i)T \)
7 \( 1 + (0.212 + 0.977i)T \)
11 \( 1 + (0.540 + 0.841i)T \)
17 \( 1 + (-0.755 + 0.654i)T \)
19 \( 1 + (0.877 - 0.479i)T \)
23 \( 1 + (-0.877 + 0.479i)T \)
29 \( 1 + (0.212 + 0.977i)T \)
31 \( 1 + (-0.479 + 0.877i)T \)
37 \( 1 + (0.707 - 0.707i)T \)
41 \( 1 + (0.599 - 0.800i)T \)
43 \( 1 + (-0.212 + 0.977i)T \)
47 \( 1 + (-0.142 - 0.989i)T \)
53 \( 1 + (-0.989 - 0.142i)T \)
59 \( 1 + (-0.800 - 0.599i)T \)
61 \( 1 + (-0.349 - 0.936i)T \)
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 \( 1 + (0.0713 - 0.997i)T \)
97 \( 1 + (-0.540 + 0.841i)T \)
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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

−20.75177388953303611396306139943, −20.18657640752264252379995674050, −19.48211480603605743119631342374, −18.49757349788329202792383113833, −17.89482116206864685008359822734, −16.98582682441112537034927390071, −16.35993344355245344339347453135, −15.89250909073070804315398699156, −15.010933001970623441318944734385, −14.30807867604061240927458765864, −13.60237729269386399904407009901, −11.84832242348449166636810310097, −11.36533052434243233229333519928, −10.71504719196968106125069743665, −9.96039884653239306093470191691, −9.171144832837855712360429103597, −8.11337263704722653138258372221, −7.45431648114578829558676514037, −6.481975024897937582350224770466, −5.901782951413747070652706613701, −4.58454457669698118722714279936, −4.07360253419586384223091924136, −2.90272367134361627715634130901, −1.10713810137383336645344050500, −0.103331047314007075199903531152, 1.36454640408492463398909446909, 1.96362975586959652263461818394, 3.15617682892281427168414505456, 4.34365629344167576449080923193, 5.186267398975245345633945441597, 6.41725930128178662919444863335, 7.313314116390513639934123751666, 7.968417521970029370581460776536, 8.81312650183304312600314395342, 9.451841684491126204241017065156, 10.76874340506104785797554020948, 11.41007940216877138556672668678, 12.14550836820540913983504415709, 12.46366048218178569899107796018, 13.26510020565535495740090538532, 14.54477365502068059410564808354, 15.72276221456041255590753798164, 16.17465241511247079517650192423, 17.19952297408701281203201076874, 17.897722876570836486009902644637, 18.282704974737959390298936336498, 19.37288112744125034125149116868, 19.76024654625734708728760722688, 20.38295489168958182565300879921, 21.67452650231793632164450083631

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