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Properties

Label 1-1157-1157.1058-r1-0-0
Degree $1$
Conductor $1157$
Sign $0.762 - 0.646i$
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  + (−0.755 + 0.654i)2-s + (−0.989 + 0.142i)3-s + (0.142 − 0.989i)4-s + (0.540 + 0.841i)5-s + (0.654 − 0.755i)6-s + (0.841 − 0.540i)7-s + (0.540 + 0.841i)8-s + (0.959 − 0.281i)9-s + (−0.959 − 0.281i)10-s + (−0.540 + 0.841i)11-s + i·12-s + (−0.281 + 0.959i)14-s + (−0.654 − 0.755i)15-s + (−0.959 − 0.281i)16-s + (−0.654 + 0.755i)17-s + (−0.540 + 0.841i)18-s + ⋯
L(s)  = 1  + (−0.755 + 0.654i)2-s + (−0.989 + 0.142i)3-s + (0.142 − 0.989i)4-s + (0.540 + 0.841i)5-s + (0.654 − 0.755i)6-s + (0.841 − 0.540i)7-s + (0.540 + 0.841i)8-s + (0.959 − 0.281i)9-s + (−0.959 − 0.281i)10-s + (−0.540 + 0.841i)11-s + i·12-s + (−0.281 + 0.959i)14-s + (−0.654 − 0.755i)15-s + (−0.959 − 0.281i)16-s + (−0.654 + 0.755i)17-s + (−0.540 + 0.841i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.762 - 0.646i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.762 - 0.646i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1157\)    =    \(13 \cdot 89\)
Sign: $0.762 - 0.646i$
Analytic conductor: \(124.336\)
Root analytic conductor: \(124.336\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1157} (1058, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1157,\ (1:\ ),\ 0.762 - 0.646i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3475726964 - 0.1275849806i\)
\(L(\frac12)\) \(\approx\) \(0.3475726964 - 0.1275849806i\)
\(L(1)\) \(\approx\) \(0.5047361231 + 0.2288105494i\)
\(L(1)\) \(\approx\) \(0.5047361231 + 0.2288105494i\)

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 \( 1 + (-0.989 + 0.142i)T \)
5 \( 1 + (0.540 + 0.841i)T \)
7 \( 1 + (0.841 - 0.540i)T \)
11 \( 1 + (-0.540 + 0.841i)T \)
17 \( 1 + (-0.654 + 0.755i)T \)
19 \( 1 + (-0.959 + 0.281i)T \)
23 \( 1 + (-0.281 - 0.959i)T \)
29 \( 1 + (0.540 + 0.841i)T \)
31 \( 1 + (-0.959 - 0.281i)T \)
37 \( 1 - T \)
41 \( 1 + (0.142 - 0.989i)T \)
43 \( 1 + (-0.540 + 0.841i)T \)
47 \( 1 + (0.989 + 0.142i)T \)
53 \( 1 + (0.142 + 0.989i)T \)
59 \( 1 + (0.142 - 0.989i)T \)
61 \( 1 + (0.909 - 0.415i)T \)
67 \( 1 + (-0.989 + 0.142i)T \)
71 \( 1 + (-0.540 + 0.841i)T \)
73 \( 1 + (0.281 - 0.959i)T \)
79 \( 1 + (0.959 + 0.281i)T \)
83 \( 1 + (0.654 - 0.755i)T \)
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

−21.28894947041482192201827250736, −20.55588600442599245478702245054, −19.50473415712159385476882995557, −18.723110623292393665980123534312, −17.84028780820233347885615750013, −17.64894749025623941380967653768, −16.72442572934920784894925150242, −16.09880925104669545191288348386, −15.33028657443349413392609867396, −13.66171780738468567112422679113, −13.23549532588311201319622212141, −12.24155723102891083101775031651, −11.68948726099256245101464795646, −10.95879566895158655586917207589, −10.247146488144604318672326639826, −9.23056437457606748596655938293, −8.56317306082731784046676972334, −7.77018034678512411526643781003, −6.68437817721837101839634498949, −5.62719781350715941075344448148, −4.96063006087503061897893668480, −4.021487021987880210271731473774, −2.464306200909048896548322787847, −1.73188578422344315335774677746, −0.75688210824762253005036532476, 0.14525246640885043650382111713, 1.55969672756464327919460603235, 2.187175467439266043848822756485, 4.08514073915092362806538772070, 4.913562681168583281833458993666, 5.754711404395680550915004217336, 6.63313703572521229142848837502, 7.12701486600339496661295488358, 8.041635488678641669571761337992, 9.101555526420011616848050399077, 10.24766023374821246583756554341, 10.57521254762650024449189353428, 11.03097341158387069691007466961, 12.25738018657222485147500089668, 13.25570378666203798899205292071, 14.34780720268170638666553891141, 14.9024095089445188563177566311, 15.618048357756459983252623143806, 16.60899170031873283250020991548, 17.359151588213353152132674464639, 17.72510898463459104307784336938, 18.36367782307769278036459935302, 19.07858674084864076925012116037, 20.25476180315703374814971755506, 20.97234816794241470101595719687

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