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Properties

Label 1-1157-1157.472-r1-0-0
Degree $1$
Conductor $1157$
Sign $0.998 - 0.0531i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.235 − 0.971i)2-s + (−0.304 − 0.952i)3-s + (−0.888 + 0.458i)4-s + (0.755 − 0.654i)5-s + (−0.853 + 0.520i)6-s + (−0.828 − 0.560i)7-s + (0.654 + 0.755i)8-s + (−0.814 + 0.580i)9-s + (−0.814 − 0.580i)10-s + (0.981 + 0.189i)11-s + (0.707 + 0.707i)12-s + (−0.349 + 0.936i)14-s + (−0.853 − 0.520i)15-s + (0.580 − 0.814i)16-s + (−0.971 − 0.235i)17-s + (0.755 + 0.654i)18-s + ⋯
L(s)  = 1  + (−0.235 − 0.971i)2-s + (−0.304 − 0.952i)3-s + (−0.888 + 0.458i)4-s + (0.755 − 0.654i)5-s + (−0.853 + 0.520i)6-s + (−0.828 − 0.560i)7-s + (0.654 + 0.755i)8-s + (−0.814 + 0.580i)9-s + (−0.814 − 0.580i)10-s + (0.981 + 0.189i)11-s + (0.707 + 0.707i)12-s + (−0.349 + 0.936i)14-s + (−0.853 − 0.520i)15-s + (0.580 − 0.814i)16-s + (−0.971 − 0.235i)17-s + (0.755 + 0.654i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.998 - 0.0531i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.998 - 0.0531i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1157\)    =    \(13 \cdot 89\)
Sign: $0.998 - 0.0531i$
Analytic conductor: \(124.336\)
Root analytic conductor: \(124.336\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1157} (472, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1157,\ (1:\ ),\ 0.998 - 0.0531i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1395123515 + 0.003707007084i\)
\(L(\frac12)\) \(\approx\) \(0.1395123515 + 0.003707007084i\)
\(L(1)\) \(\approx\) \(0.4280937068 - 0.5601676636i\)
\(L(1)\) \(\approx\) \(0.4280937068 - 0.5601676636i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
89 \( 1 \)
good2 \( 1 + (-0.235 - 0.971i)T \)
3 \( 1 + (-0.304 - 0.952i)T \)
5 \( 1 + (0.755 - 0.654i)T \)
7 \( 1 + (-0.828 - 0.560i)T \)
11 \( 1 + (0.981 + 0.189i)T \)
17 \( 1 + (-0.971 - 0.235i)T \)
19 \( 1 + (0.986 + 0.165i)T \)
23 \( 1 + (-0.771 - 0.636i)T \)
29 \( 1 + (-0.436 - 0.899i)T \)
31 \( 1 + (-0.936 - 0.349i)T \)
37 \( 1 + (-0.965 - 0.258i)T \)
41 \( 1 + (0.739 - 0.672i)T \)
43 \( 1 + (-0.436 + 0.899i)T \)
47 \( 1 + (-0.540 + 0.841i)T \)
53 \( 1 + (-0.540 - 0.841i)T \)
59 \( 1 + (-0.304 + 0.952i)T \)
61 \( 1 + (-0.393 - 0.919i)T \)
67 \( 1 + (0.888 + 0.458i)T \)
71 \( 1 + (0.945 - 0.327i)T \)
73 \( 1 + (0.415 + 0.909i)T \)
79 \( 1 + (-0.909 + 0.415i)T \)
83 \( 1 + (-0.877 - 0.479i)T \)
97 \( 1 + (-0.981 + 0.189i)T \)
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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

−21.688724980580855682115319479821, −20.16274095169353984832525062921, −19.53365674111997755635341328618, −18.43186824755367353209530649214, −17.89392963572652921769443829383, −17.09708972564706295977427423230, −16.43059663819055718815038112653, −15.66437736847280745471242638325, −15.09671883055788410173165679129, −14.23894197888197980785610215603, −13.68337232230650592870279234803, −12.57901426190228026967589343551, −11.443563874528830320119532611847, −10.57505939465297811346236100176, −9.67956177509100834528972197249, −9.298824020872957062135851380862, −8.602617189234028250693583830125, −7.10335987916101506147457695998, −6.47388283996615270304384116193, −5.762965777223462146016673875596, −5.12780995022057808707530314837, −3.85380271890435101807690770323, −3.19350235009348839436065239857, −1.66838703663772495106947564568, −0.03987950608019390278223995682, 0.82367863836223393324282229735, 1.69094370964711938338341465939, 2.473857370436410727021419433758, 3.67237388965847989776705964247, 4.62126349220389023551614875352, 5.70980169354111971227921530015, 6.54070556013501940682549052675, 7.47161400836721785717672785863, 8.49150940292106755541471933401, 9.35569465947104249927173106049, 9.8613143867480688244309920468, 10.97034190656586070023062685160, 11.71187172869766569685006949810, 12.61182433645477285116421070179, 12.95690601526493831636648886975, 13.893505134574055132071189788651, 14.1953952053522324637195937940, 16.059742525236749414810229921621, 16.80131177296452518843817526630, 17.40164349119181513213331496122, 18.012588179954827161609794659619, 18.81233501833024818869449286618, 19.774617152663653676108346912362, 20.03866850819135518042267598303, 20.82697951251130246780793006168

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