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Properties

Label 1-115-115.7-r0-0-0
Degree $1$
Conductor $115$
Sign $0.376 - 0.926i$
Analytic cond. $0.534057$
Root an. cond. $0.534057$
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  + (0.989 − 0.142i)2-s + (−0.909 − 0.415i)3-s + (0.959 − 0.281i)4-s + (−0.959 − 0.281i)6-s + (−0.540 − 0.841i)7-s + (0.909 − 0.415i)8-s + (0.654 + 0.755i)9-s + (0.142 − 0.989i)11-s + (−0.989 − 0.142i)12-s + (0.540 − 0.841i)13-s + (−0.654 − 0.755i)14-s + (0.841 − 0.540i)16-s + (−0.281 + 0.959i)17-s + (0.755 + 0.654i)18-s + (−0.959 + 0.281i)19-s + ⋯
L(s)  = 1  + (0.989 − 0.142i)2-s + (−0.909 − 0.415i)3-s + (0.959 − 0.281i)4-s + (−0.959 − 0.281i)6-s + (−0.540 − 0.841i)7-s + (0.909 − 0.415i)8-s + (0.654 + 0.755i)9-s + (0.142 − 0.989i)11-s + (−0.989 − 0.142i)12-s + (0.540 − 0.841i)13-s + (−0.654 − 0.755i)14-s + (0.841 − 0.540i)16-s + (−0.281 + 0.959i)17-s + (0.755 + 0.654i)18-s + (−0.959 + 0.281i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.376 - 0.926i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.376 - 0.926i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(115\)    =    \(5 \cdot 23\)
Sign: $0.376 - 0.926i$
Analytic conductor: \(0.534057\)
Root analytic conductor: \(0.534057\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{115} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 115,\ (0:\ ),\ 0.376 - 0.926i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.155983455 - 0.7782059310i\)
\(L(\frac12)\) \(\approx\) \(1.155983455 - 0.7782059310i\)
\(L(1)\) \(\approx\) \(1.284571036 - 0.4869193474i\)
\(L(1)\) \(\approx\) \(1.284571036 - 0.4869193474i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
23 \( 1 \)
good2 \( 1 + (0.989 - 0.142i)T \)
3 \( 1 + (-0.909 - 0.415i)T \)
7 \( 1 + (-0.540 - 0.841i)T \)
11 \( 1 + (0.142 - 0.989i)T \)
13 \( 1 + (0.540 - 0.841i)T \)
17 \( 1 + (-0.281 + 0.959i)T \)
19 \( 1 + (-0.959 + 0.281i)T \)
29 \( 1 + (0.959 + 0.281i)T \)
31 \( 1 + (0.415 + 0.909i)T \)
37 \( 1 + (-0.755 + 0.654i)T \)
41 \( 1 + (-0.654 + 0.755i)T \)
43 \( 1 + (0.909 + 0.415i)T \)
47 \( 1 + iT \)
53 \( 1 + (0.540 + 0.841i)T \)
59 \( 1 + (-0.841 - 0.540i)T \)
61 \( 1 + (-0.415 - 0.909i)T \)
67 \( 1 + (-0.989 + 0.142i)T \)
71 \( 1 + (-0.142 - 0.989i)T \)
73 \( 1 + (0.281 + 0.959i)T \)
79 \( 1 + (0.841 + 0.540i)T \)
83 \( 1 + (0.755 - 0.654i)T \)
89 \( 1 + (0.415 - 0.909i)T \)
97 \( 1 + (0.755 + 0.654i)T \)
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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

−29.38098105454142628787105501632, −28.61944028760562787737984609036, −27.7924650855367395974859138196, −26.20474204508292732917853425726, −25.28330545335047595175044819248, −24.14423435934477817064257487260, −23.072617485367417159416083536175, −22.49176431884656747442525416421, −21.49508082017688557718907970517, −20.70482877746538467236212194199, −19.21953227482288810108535315024, −17.84902784765237986348097604608, −16.66035035643467890421525339585, −15.72818761449931917421838733197, −15.03077364538847602600803418639, −13.516406012834665936865592994288, −12.29577572803611638152284159008, −11.712382184622862796418017546644, −10.422314478836245812708736740543, −9.06953143398937680412114079695, −7.00540539941407201906361432605, −6.20258572403455746774233067131, −4.99228380728934958326124257602, −3.97680820904736379960508263181, −2.24348010720837700191310725636, 1.25346568363340372755584521850, 3.25303703833377321323337793894, 4.53897640469440678116152914171, 5.99710894669572728646829103163, 6.5939662315411755129833012499, 8.06319624113299155626193091041, 10.45353783629957900730389938476, 10.86463670043001871542655266822, 12.29087981030251670488160513352, 13.11616324455739536336930698643, 13.96957959281231903121799024354, 15.54882201604841487716667686257, 16.47967066408680154481127733915, 17.39984320306094851710638766274, 18.9954644989806739023204968026, 19.80388854489936848048090530314, 21.19904335514150077936835503877, 22.10402036211174791503692009510, 23.10651970536787162111588572379, 23.65187260812023299915350370983, 24.679728379965628463812208524696, 25.781599759563436414050415336468, 27.27923274525852663004347430668, 28.45046629317731169417977369001, 29.40426527940913683307469103066

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