Properties

Label 1-5e3-125.19-r0-0-0
Degree $1$
Conductor $125$
Sign $-0.998 + 0.0502i$
Analytic cond. $0.580497$
Root an. cond. $0.580497$
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  + (0.425 + 0.904i)2-s + (−0.0627 + 0.998i)3-s + (−0.637 + 0.770i)4-s + (−0.929 + 0.368i)6-s + (−0.309 + 0.951i)7-s + (−0.968 − 0.248i)8-s + (−0.992 − 0.125i)9-s + (−0.425 − 0.904i)11-s + (−0.728 − 0.684i)12-s + (0.992 + 0.125i)13-s + (−0.992 + 0.125i)14-s + (−0.187 − 0.982i)16-s + (0.637 + 0.770i)17-s + (−0.309 − 0.951i)18-s + (0.0627 + 0.998i)19-s + ⋯
L(s)  = 1  + (0.425 + 0.904i)2-s + (−0.0627 + 0.998i)3-s + (−0.637 + 0.770i)4-s + (−0.929 + 0.368i)6-s + (−0.309 + 0.951i)7-s + (−0.968 − 0.248i)8-s + (−0.992 − 0.125i)9-s + (−0.425 − 0.904i)11-s + (−0.728 − 0.684i)12-s + (0.992 + 0.125i)13-s + (−0.992 + 0.125i)14-s + (−0.187 − 0.982i)16-s + (0.637 + 0.770i)17-s + (−0.309 − 0.951i)18-s + (0.0627 + 0.998i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 125 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0502i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 125 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0502i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(125\)    =    \(5^{3}\)
Sign: $-0.998 + 0.0502i$
Analytic conductor: \(0.580497\)
Root analytic conductor: \(0.580497\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{125} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 125,\ (0:\ ),\ -0.998 + 0.0502i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.02593268249 + 1.031611389i\)
\(L(\frac12)\) \(\approx\) \(0.02593268249 + 1.031611389i\)
\(L(1)\) \(\approx\) \(0.5958900380 + 0.8822730763i\)
\(L(1)\) \(\approx\) \(0.5958900380 + 0.8822730763i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
good2 \( 1 + (0.425 + 0.904i)T \)
3 \( 1 + (-0.0627 + 0.998i)T \)
7 \( 1 + (-0.309 + 0.951i)T \)
11 \( 1 + (-0.425 - 0.904i)T \)
13 \( 1 + (0.992 + 0.125i)T \)
17 \( 1 + (0.637 + 0.770i)T \)
19 \( 1 + (0.0627 + 0.998i)T \)
23 \( 1 + (-0.876 - 0.481i)T \)
29 \( 1 + (0.535 + 0.844i)T \)
31 \( 1 + (-0.637 - 0.770i)T \)
37 \( 1 + (0.187 + 0.982i)T \)
41 \( 1 + (0.876 - 0.481i)T \)
43 \( 1 + (0.809 + 0.587i)T \)
47 \( 1 + (-0.968 + 0.248i)T \)
53 \( 1 + (0.929 + 0.368i)T \)
59 \( 1 + (0.728 + 0.684i)T \)
61 \( 1 + (0.876 + 0.481i)T \)
67 \( 1 + (-0.535 + 0.844i)T \)
71 \( 1 + (0.968 - 0.248i)T \)
73 \( 1 + (-0.728 + 0.684i)T \)
79 \( 1 + (0.0627 - 0.998i)T \)
83 \( 1 + (-0.0627 - 0.998i)T \)
89 \( 1 + (0.728 - 0.684i)T \)
97 \( 1 + (-0.535 - 0.844i)T \)
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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

−28.62603734252733865585977739154, −27.85049507186575448408593398468, −26.37217456884153851788076993664, −25.36671476348725973545425110957, −23.96383437438642246874382861316, −23.22828336554172819054084521839, −22.67392594009520946353332655671, −21.09303631280732455168788878737, −20.12989335703127647735191022424, −19.47395402016753109128992177883, −18.20610627776392342745984744609, −17.61467950334755827067072883034, −15.942340849016031529150454776581, −14.34473866034581163347427716666, −13.508118677225205310286905397550, −12.76783502789384585689483780967, −11.6477999245048671061420289274, −10.61513564506624244953001947729, −9.3970213386925970262287279015, −7.82904887226903989616713270422, −6.61349714628697256872555835329, −5.24853559917938827621666132979, −3.73619480686890788191466485464, −2.35280003364568684221290676905, −0.887857656936832619916044497070, 3.057226800601453913289280737705, 4.07157686785020646693928173755, 5.651626007646838249501265562131, 6.05442677279285443609242519087, 8.15881633544301335417370874026, 8.83136545354960794541398460676, 10.16006487503325358167038398759, 11.597648987239847888522815253334, 12.80350976930608106630914747013, 14.13917072196322506166091447124, 14.996680581687088768739256441395, 16.154829266560665678110541031932, 16.406747678023889531974158871204, 17.96414516723501406332914110507, 18.95835983683041203275263537974, 20.769688070625839539442476488557, 21.51590817117196845078860955617, 22.30833579094618414260886235260, 23.292569106571360228948685492344, 24.35863287260889643773497696964, 25.672575281342217329960572671734, 26.05149530767384073357851919553, 27.33169494608014315096288105005, 28.05276996130186240498676975426, 29.29795227493933522437022831078

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