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Properties

Label 1-10e2-100.91-r1-0-0
Degree $1$
Conductor $100$
Sign $-0.425 - 0.904i$
Analytic cond. $10.7464$
Root an. cond. $10.7464$
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  + (0.809 − 0.587i)3-s − 7-s + (0.309 − 0.951i)9-s + (−0.309 − 0.951i)11-s + (0.309 − 0.951i)13-s + (−0.809 − 0.587i)17-s + (0.809 + 0.587i)19-s + (−0.809 + 0.587i)21-s + (−0.309 − 0.951i)23-s + (−0.309 − 0.951i)27-s + (−0.809 + 0.587i)29-s + (0.809 + 0.587i)31-s + (−0.809 − 0.587i)33-s + (0.309 − 0.951i)37-s + (−0.309 − 0.951i)39-s + ⋯
L(s)  = 1  + (0.809 − 0.587i)3-s − 7-s + (0.309 − 0.951i)9-s + (−0.309 − 0.951i)11-s + (0.309 − 0.951i)13-s + (−0.809 − 0.587i)17-s + (0.809 + 0.587i)19-s + (−0.809 + 0.587i)21-s + (−0.309 − 0.951i)23-s + (−0.309 − 0.951i)27-s + (−0.809 + 0.587i)29-s + (0.809 + 0.587i)31-s + (−0.809 − 0.587i)33-s + (0.309 − 0.951i)37-s + (−0.309 − 0.951i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.425 - 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.425 - 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(100\)    =    \(2^{2} \cdot 5^{2}\)
Sign: $-0.425 - 0.904i$
Analytic conductor: \(10.7464\)
Root analytic conductor: \(10.7464\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{100} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 100,\ (1:\ ),\ -0.425 - 0.904i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8766659312 - 1.381404465i\)
\(L(\frac12)\) \(\approx\) \(0.8766659312 - 1.381404465i\)
\(L(1)\) \(\approx\) \(1.081388548 - 0.5088628249i\)
\(L(1)\) \(\approx\) \(1.081388548 - 0.5088628249i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
good3 \( 1 + (0.809 - 0.587i)T \)
7 \( 1 - T \)
11 \( 1 + (-0.309 - 0.951i)T \)
13 \( 1 + (0.309 - 0.951i)T \)
17 \( 1 + (-0.809 - 0.587i)T \)
19 \( 1 + (0.809 + 0.587i)T \)
23 \( 1 + (-0.309 - 0.951i)T \)
29 \( 1 + (-0.809 + 0.587i)T \)
31 \( 1 + (0.809 + 0.587i)T \)
37 \( 1 + (0.309 - 0.951i)T \)
41 \( 1 + (0.309 - 0.951i)T \)
43 \( 1 - T \)
47 \( 1 + (0.809 - 0.587i)T \)
53 \( 1 + (-0.809 + 0.587i)T \)
59 \( 1 + (-0.309 + 0.951i)T \)
61 \( 1 + (0.309 + 0.951i)T \)
67 \( 1 + (0.809 + 0.587i)T \)
71 \( 1 + (0.809 - 0.587i)T \)
73 \( 1 + (0.309 + 0.951i)T \)
79 \( 1 + (0.809 - 0.587i)T \)
83 \( 1 + (0.809 + 0.587i)T \)
89 \( 1 + (0.309 + 0.951i)T \)
97 \( 1 + (-0.809 + 0.587i)T \)
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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

−30.24615714896790073330705470208, −28.74431315818797254959667785543, −28.08373105489283670368226275825, −26.53137720985088388777766558092, −26.08803405214710040190901032169, −25.12369461741461590002489448169, −23.81812575852650052871137352290, −22.50956893417377456953233178968, −21.65999650981566356339488933406, −20.4480744630058923183436069852, −19.65527082418544219971175256652, −18.63529763095557945643152890254, −17.10740479751194009774167731158, −15.84984227122171565308816471233, −15.22930769967971878173273303110, −13.78784386483765987010101211287, −12.980353061389130288440278164480, −11.37097444406704119480845023184, −9.868203450849767536774734686318, −9.30204804026100103237633552484, −7.828921442575390296806767069154, −6.507893844168187548348781212775, −4.70779111615043734242485773058, −3.520180667568283749031222535, −2.11339860590120723361623431921, 0.64137602175394656600372314701, 2.63975559294522670082553493352, 3.6278295194698934810972369339, 5.76120343778232217872878691512, 6.97226152605682535644106722173, 8.231898837148160993256276256272, 9.26916468857723266957295390019, 10.58174974812249823811777080684, 12.23153643508396595840924423055, 13.22079360501403320305683916530, 14.02133146096858480011795136366, 15.44460274203615558660128542124, 16.34985778180431588617065991601, 18.02695019581774911426945875832, 18.80243520614971783169961517363, 19.88385278796923605576659389532, 20.66149888086729297366626143673, 22.12063142293760181826736299106, 23.1418415768954438213430350305, 24.43765355289426877635143857338, 25.12645808754741957099875940935, 26.262641828281791479616076171637, 26.94789320567263350275019167036, 28.61410971977832494221238475951, 29.460700846449877005306814306095

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