Properties

Label 1-115-115.12-r1-0-0
Degree $1$
Conductor $115$
Sign $-0.996 - 0.0845i$
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  + (0.909 + 0.415i)2-s + (−0.281 + 0.959i)3-s + (0.654 + 0.755i)4-s + (−0.654 + 0.755i)6-s + (−0.989 − 0.142i)7-s + (0.281 + 0.959i)8-s + (−0.841 − 0.540i)9-s + (0.415 + 0.909i)11-s + (−0.909 + 0.415i)12-s + (−0.989 + 0.142i)13-s + (−0.841 − 0.540i)14-s + (−0.142 + 0.989i)16-s + (−0.755 − 0.654i)17-s + (−0.540 − 0.841i)18-s + (0.654 + 0.755i)19-s + ⋯
L(s)  = 1  + (0.909 + 0.415i)2-s + (−0.281 + 0.959i)3-s + (0.654 + 0.755i)4-s + (−0.654 + 0.755i)6-s + (−0.989 − 0.142i)7-s + (0.281 + 0.959i)8-s + (−0.841 − 0.540i)9-s + (0.415 + 0.909i)11-s + (−0.909 + 0.415i)12-s + (−0.989 + 0.142i)13-s + (−0.841 − 0.540i)14-s + (−0.142 + 0.989i)16-s + (−0.755 − 0.654i)17-s + (−0.540 − 0.841i)18-s + (0.654 + 0.755i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.996 - 0.0845i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.996 - 0.0845i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(115\)    =    \(5 \cdot 23\)
Sign: $-0.996 - 0.0845i$
Analytic conductor: \(12.3584\)
Root analytic conductor: \(12.3584\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{115} (12, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 115,\ (1:\ ),\ -0.996 - 0.0845i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.07109228344 + 1.677856383i\)
\(L(\frac12)\) \(\approx\) \(-0.07109228344 + 1.677856383i\)
\(L(1)\) \(\approx\) \(0.9311879202 + 0.9229494465i\)
\(L(1)\) \(\approx\) \(0.9311879202 + 0.9229494465i\)

Euler product

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

−29.06202413438570368690218199926, −27.95103562233025577099509840379, −26.30249275995827202816446796647, −24.98556999238115666460800273701, −24.33336933084853235211982832042, −23.39621895353033191597499959754, −22.25365808152142611269426962316, −21.81506644338489528988924796537, −19.83740880491786774593009096038, −19.59279204503663133384034761288, −18.412471586709984214270520150579, −16.955137708274637112591005951146, −15.84204926078812993392710566943, −14.45868427876864109149241528181, −13.42338386466081512889900238575, −12.65013196078732911659984481547, −11.704804236947307109647455630184, −10.59633569866255802001989852297, −9.05105612696561701785359138981, −7.20304038172538642384210211338, −6.32245299582408540370077758433, −5.24747293871594010283543915442, −3.44116626244563594729154666911, −2.24092777138098893162271281768, −0.50571762023202468569608423831, 2.70627280273594073154454121643, 3.98223503979501127304532228317, 4.96578853601680047067123715615, 6.25712797724178380137060530066, 7.3518688345690150140842819710, 9.2006933180062952293800223946, 10.20598420840010938857169943366, 11.68411752322292677837647503666, 12.523335892085287530400084495650, 13.93386754552253295421343643785, 14.94956904649015088136131381572, 15.87692770607625431183554879458, 16.68835546664336944819599101049, 17.67367088068933239011025442183, 19.72112159833573661281953019716, 20.47023915451802958165308979785, 21.672378763999730066196881376977, 22.60838663264258432230819234538, 22.92473395482703045420983683825, 24.44631180050633292633836060906, 25.49715591708211938629846284644, 26.380639912370305333177258636545, 27.33268253590942522451372141906, 28.861707048993244129026515399322, 29.36425519973831640337082809744

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