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Properties

Label 1-1157-1157.564-r0-0-0
Degree $1$
Conductor $1157$
Sign $0.627 + 0.778i$
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  + (−0.909 − 0.415i)2-s + (0.997 − 0.0713i)3-s + (0.654 + 0.755i)4-s + (0.959 − 0.281i)5-s + (−0.936 − 0.349i)6-s + (−0.479 + 0.877i)7-s + (−0.281 − 0.959i)8-s + (0.989 − 0.142i)9-s + (−0.989 − 0.142i)10-s + (−0.281 + 0.959i)11-s + (0.707 + 0.707i)12-s + (0.800 − 0.599i)14-s + (0.936 − 0.349i)15-s + (−0.142 + 0.989i)16-s + (−0.909 + 0.415i)17-s + (−0.959 − 0.281i)18-s + ⋯
L(s)  = 1  + (−0.909 − 0.415i)2-s + (0.997 − 0.0713i)3-s + (0.654 + 0.755i)4-s + (0.959 − 0.281i)5-s + (−0.936 − 0.349i)6-s + (−0.479 + 0.877i)7-s + (−0.281 − 0.959i)8-s + (0.989 − 0.142i)9-s + (−0.989 − 0.142i)10-s + (−0.281 + 0.959i)11-s + (0.707 + 0.707i)12-s + (0.800 − 0.599i)14-s + (0.936 − 0.349i)15-s + (−0.142 + 0.989i)16-s + (−0.909 + 0.415i)17-s + (−0.959 − 0.281i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.627 + 0.778i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.627 + 0.778i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1157\)    =    \(13 \cdot 89\)
Sign: $0.627 + 0.778i$
Analytic conductor: \(5.37308\)
Root analytic conductor: \(5.37308\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1157} (564, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1157,\ (0:\ ),\ 0.627 + 0.778i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.319503487 + 0.6314087369i\)
\(L(\frac12)\) \(\approx\) \(1.319503487 + 0.6314087369i\)
\(L(1)\) \(\approx\) \(1.072700456 + 0.07123084408i\)
\(L(1)\) \(\approx\) \(1.072700456 + 0.07123084408i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
89 \( 1 \)
good2 \( 1 + (-0.909 - 0.415i)T \)
3 \( 1 + (0.997 - 0.0713i)T \)
5 \( 1 + (0.959 - 0.281i)T \)
7 \( 1 + (-0.479 + 0.877i)T \)
11 \( 1 + (-0.281 + 0.959i)T \)
17 \( 1 + (-0.909 + 0.415i)T \)
19 \( 1 + (-0.599 + 0.800i)T \)
23 \( 1 + (0.599 - 0.800i)T \)
29 \( 1 + (-0.479 + 0.877i)T \)
31 \( 1 + (-0.800 + 0.599i)T \)
37 \( 1 + (0.707 + 0.707i)T \)
41 \( 1 + (-0.997 - 0.0713i)T \)
43 \( 1 + (0.479 + 0.877i)T \)
47 \( 1 + (-0.654 - 0.755i)T \)
53 \( 1 + (0.755 + 0.654i)T \)
59 \( 1 + (0.0713 - 0.997i)T \)
61 \( 1 + (0.212 + 0.977i)T \)
67 \( 1 + (0.755 + 0.654i)T \)
71 \( 1 + (0.959 + 0.281i)T \)
73 \( 1 + (0.989 + 0.142i)T \)
79 \( 1 + (-0.989 - 0.142i)T \)
83 \( 1 + (0.936 + 0.349i)T \)
97 \( 1 + (0.281 + 0.959i)T \)
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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

−21.0077039214334532076100904441, −20.19688980671544074978945362165, −19.557698183502299216847633895, −18.8718908868310453621301714954, −18.16617496838414748980240103458, −17.27463902743888157377462074236, −16.64902935522868383864145473988, −15.7464749728866417359874595564, −15.07377930538335699456769076089, −14.15937544178350749638929095614, −13.48806564618520000199359758978, −13.05249037918364004432525667508, −11.177928779088376030836508310175, −10.75300121344079677036560087782, −9.74500183418730072206669191429, −9.30856934987660403417767084133, −8.53808461230083012806537317057, −7.52488568596907377211944518050, −6.884690075742084379561696368665, −6.106336658034306889577462417722, −5.001126017982927304206162852084, −3.67240816472859504703416602404, −2.66004388007914228720803966049, −1.94127231303402406131372497876, −0.66714492726856002635216113473, 1.46725156046881832918495264987, 2.16162097529945802324064628653, 2.74128576977511773942704409503, 3.86629870623281856622936417614, 5.06339149653762175757834348452, 6.418007690224871702185880265505, 6.95439214713977958440351875391, 8.20373568782719711388558353988, 8.76296581389013635695509678189, 9.41850219714282155454321395895, 10.04932810262516361966340343249, 10.79069699857837866937190645938, 12.2070708036844598396589589996, 12.834030234075571573546964256494, 13.18811096470917909908342031526, 14.61825025757377335443194931380, 15.12253829567334218600532167071, 16.085931174298775686239706514594, 16.83326556965836157090237272855, 17.829942896176136420482265602082, 18.41361546425300791643006521210, 18.945701110421471527920246990390, 20.085908414783325929627807285629, 20.28473826717057774565616282397, 21.33887503830613288421887930425

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