Properties

Label 1-115-115.32-r1-0-0
Degree $1$
Conductor $115$
Sign $-0.101 + 0.994i$
Analytic cond. $12.3584$
Root an. cond. $12.3584$
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  + (0.540 + 0.841i)2-s + (0.989 + 0.142i)3-s + (−0.415 + 0.909i)4-s + (0.415 + 0.909i)6-s + (0.755 − 0.654i)7-s + (−0.989 + 0.142i)8-s + (0.959 + 0.281i)9-s + (0.841 + 0.540i)11-s + (−0.540 + 0.841i)12-s + (0.755 + 0.654i)13-s + (0.959 + 0.281i)14-s + (−0.654 − 0.755i)16-s + (−0.909 + 0.415i)17-s + (0.281 + 0.959i)18-s + (−0.415 + 0.909i)19-s + ⋯
L(s)  = 1  + (0.540 + 0.841i)2-s + (0.989 + 0.142i)3-s + (−0.415 + 0.909i)4-s + (0.415 + 0.909i)6-s + (0.755 − 0.654i)7-s + (−0.989 + 0.142i)8-s + (0.959 + 0.281i)9-s + (0.841 + 0.540i)11-s + (−0.540 + 0.841i)12-s + (0.755 + 0.654i)13-s + (0.959 + 0.281i)14-s + (−0.654 − 0.755i)16-s + (−0.909 + 0.415i)17-s + (0.281 + 0.959i)18-s + (−0.415 + 0.909i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.101 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.101 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(115\)    =    \(5 \cdot 23\)
Sign: $-0.101 + 0.994i$
Analytic conductor: \(12.3584\)
Root analytic conductor: \(12.3584\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{115} (32, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 115,\ (1:\ ),\ -0.101 + 0.994i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.258448834 + 2.501675482i\)
\(L(\frac12)\) \(\approx\) \(2.258448834 + 2.501675482i\)
\(L(1)\) \(\approx\) \(1.706784585 + 1.106382069i\)
\(L(1)\) \(\approx\) \(1.706784585 + 1.106382069i\)

Euler product

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

−28.94017276481996221714540724401, −27.66535997662587243179756537548, −27.09057658947661733526098881347, −25.553226549084925931142764280454, −24.54321462414681192934443735310, −23.81717947514352432126289548877, −22.25289773517225334968078603248, −21.498448886523960973342989750807, −20.47749000762355749012119914452, −19.725032126531049281409046739119, −18.647378523659189372172669357747, −17.80586508267680431326269614437, −15.66692107944939892642537308788, −14.82344490006350713569960525848, −13.86152069308602193735790477492, −12.968419674179207605845888766444, −11.71601682185934532096795815052, −10.6861636060174824921709333897, −9.100579308679853846183263590381, −8.53563558210233193949576079030, −6.62789423653162934340663907638, −5.059223490469421991626715895488, −3.731679883384624671579740063955, −2.545244637160982823960581689724, −1.267591019252301479573348953645, 1.90256197441717797083827263132, 3.87785926249197337156790678445, 4.39229578203064650725705493152, 6.30959990665502515347993373736, 7.48874566391019047004414337893, 8.43608767206406195201438520803, 9.50730280320474121617464607300, 11.224253318793056526956868473983, 12.7443548190745328725921990894, 13.81833473804989171511705919709, 14.51193984735900281478243068384, 15.403384278393596948102980928606, 16.63530665850209375961870971040, 17.640439783628790397297666750782, 18.94494578282366141116815979467, 20.36306310122952438566925703539, 21.04777187340088616911180117441, 22.178029963269992733161970048370, 23.40938028089395636754388381207, 24.34693081485947129148722783686, 25.17003221056496404608410155361, 26.16663703513649905867853478612, 26.89592277305982925530152082113, 27.89598010043959300792865971301, 29.856943837972436471437846286556

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