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Properties

Label 1-1157-1157.327-r0-0-0
Degree $1$
Conductor $1157$
Sign $0.951 + 0.307i$
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  + (0.371 − 0.928i)2-s + (−0.999 − 0.0237i)3-s + (−0.723 − 0.690i)4-s + (−0.415 − 0.909i)5-s + (−0.393 + 0.919i)6-s + (−0.165 − 0.986i)7-s + (−0.909 + 0.415i)8-s + (0.998 + 0.0475i)9-s + (−0.998 + 0.0475i)10-s + (0.814 − 0.580i)11-s + (0.707 + 0.707i)12-s + (−0.977 − 0.212i)14-s + (0.393 + 0.919i)15-s + (0.0475 + 0.998i)16-s + (0.371 + 0.928i)17-s + (0.415 − 0.909i)18-s + ⋯
L(s)  = 1  + (0.371 − 0.928i)2-s + (−0.999 − 0.0237i)3-s + (−0.723 − 0.690i)4-s + (−0.415 − 0.909i)5-s + (−0.393 + 0.919i)6-s + (−0.165 − 0.986i)7-s + (−0.909 + 0.415i)8-s + (0.998 + 0.0475i)9-s + (−0.998 + 0.0475i)10-s + (0.814 − 0.580i)11-s + (0.707 + 0.707i)12-s + (−0.977 − 0.212i)14-s + (0.393 + 0.919i)15-s + (0.0475 + 0.998i)16-s + (0.371 + 0.928i)17-s + (0.415 − 0.909i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.951 + 0.307i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.951 + 0.307i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1157\)    =    \(13 \cdot 89\)
Sign: $0.951 + 0.307i$
Analytic conductor: \(5.37308\)
Root analytic conductor: \(5.37308\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1157} (327, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1157,\ (0:\ ),\ 0.951 + 0.307i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2277190072 + 0.03582664950i\)
\(L(\frac12)\) \(\approx\) \(0.2277190072 + 0.03582664950i\)
\(L(1)\) \(\approx\) \(0.4949759420 - 0.4332851999i\)
\(L(1)\) \(\approx\) \(0.4949759420 - 0.4332851999i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
89 \( 1 \)
good2 \( 1 + (0.371 - 0.928i)T \)
3 \( 1 + (-0.999 - 0.0237i)T \)
5 \( 1 + (-0.415 - 0.909i)T \)
7 \( 1 + (-0.165 - 0.986i)T \)
11 \( 1 + (0.814 - 0.580i)T \)
17 \( 1 + (0.371 + 0.928i)T \)
19 \( 1 + (-0.739 + 0.672i)T \)
23 \( 1 + (-0.952 - 0.304i)T \)
29 \( 1 + (-0.771 + 0.636i)T \)
31 \( 1 + (0.977 + 0.212i)T \)
37 \( 1 + (-0.965 + 0.258i)T \)
41 \( 1 + (-0.520 - 0.853i)T \)
43 \( 1 + (0.771 + 0.636i)T \)
47 \( 1 + (-0.959 + 0.281i)T \)
53 \( 1 + (-0.281 + 0.959i)T \)
59 \( 1 + (0.0237 + 0.999i)T \)
61 \( 1 + (-0.899 + 0.436i)T \)
67 \( 1 + (-0.690 - 0.723i)T \)
71 \( 1 + (0.995 - 0.0950i)T \)
73 \( 1 + (-0.540 - 0.841i)T \)
79 \( 1 + (0.540 + 0.841i)T \)
83 \( 1 + (0.599 + 0.800i)T \)
97 \( 1 + (-0.814 - 0.580i)T \)
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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

−21.80031951746770713380002435351, −20.786098557693690338999682323881, −19.29565628505614794616069648127, −18.73979595904911357976571551557, −17.889621608461530589725558224344, −17.473547239417827342224290985504, −16.47366382234772210240736870866, −15.69048822548667723817358312080, −15.24022279436914404906753578403, −14.479746878948651978852355299063, −13.51846881125855021598015771719, −12.48777782395147143051225052611, −11.85774963020018502708236977206, −11.36055966190839801133920314746, −9.997595963074445469510509557898, −9.39195147661831514807794133411, −8.2117639400375571120772875903, −7.27463296258755969751763272314, −6.588739119465274074437291516877, −6.045960030405037243897240895285, −5.08603135317964539292839046962, −4.261578999581374234485635834574, −3.32938876317705899964598755083, −2.09998252611606457260075624427, −0.10961966863226431844967997924, 1.09729825783577248834930942737, 1.65553664260320811069551402523, 3.56445802440221315138180945044, 4.05950205670751324524176820945, 4.7774742004600267305572341474, 5.83074564760551025808308625548, 6.45281755529500199469233950396, 7.78864818437548979480799025591, 8.71092869177797443451001760926, 9.72083405088200860258214951060, 10.522518543174229595478455790050, 11.05574300504493834900640468963, 12.17635352793056227676081503799, 12.31793610103205087240563544333, 13.30175964212519785111281977441, 14.01113007828663450688786468275, 15.043508076719273173244206798721, 16.11505758416563166477035253997, 16.88924911247397031503238586830, 17.25969872569210813797055826442, 18.34322342992452422375340268442, 19.38356082287598750313235413774, 19.5983324154334046000005266367, 20.77525545381118542091802779324, 21.15301151063625341596480181373

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