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Properties

Label 1-1375-1375.1104-r0-0-0
Degree $1$
Conductor $1375$
Sign $-0.491 + 0.870i$
Analytic cond. $6.38547$
Root an. cond. $6.38547$
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  + (−0.728 − 0.684i)2-s + (−0.968 − 0.248i)3-s + (0.0627 + 0.998i)4-s + (0.535 + 0.844i)6-s + (0.809 + 0.587i)7-s + (0.637 − 0.770i)8-s + (0.876 + 0.481i)9-s + (0.187 − 0.982i)12-s + (−0.876 − 0.481i)13-s + (−0.187 − 0.982i)14-s + (−0.992 + 0.125i)16-s + (−0.968 + 0.248i)17-s + (−0.309 − 0.951i)18-s + (0.535 + 0.844i)19-s + (−0.637 − 0.770i)21-s + ⋯
L(s)  = 1  + (−0.728 − 0.684i)2-s + (−0.968 − 0.248i)3-s + (0.0627 + 0.998i)4-s + (0.535 + 0.844i)6-s + (0.809 + 0.587i)7-s + (0.637 − 0.770i)8-s + (0.876 + 0.481i)9-s + (0.187 − 0.982i)12-s + (−0.876 − 0.481i)13-s + (−0.187 − 0.982i)14-s + (−0.992 + 0.125i)16-s + (−0.968 + 0.248i)17-s + (−0.309 − 0.951i)18-s + (0.535 + 0.844i)19-s + (−0.637 − 0.770i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.491 + 0.870i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.491 + 0.870i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1375\)    =    \(5^{3} \cdot 11\)
Sign: $-0.491 + 0.870i$
Analytic conductor: \(6.38547\)
Root analytic conductor: \(6.38547\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1375} (1104, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1375,\ (0:\ ),\ -0.491 + 0.870i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1194310273 + 0.2045543785i\)
\(L(\frac12)\) \(\approx\) \(0.1194310273 + 0.2045543785i\)
\(L(1)\) \(\approx\) \(0.4976505415 - 0.07078071710i\)
\(L(1)\) \(\approx\) \(0.4976505415 - 0.07078071710i\)

Euler product

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

−20.53434757502478268099506651702, −19.72209805443382064628163821904, −18.83992175791048117498710234989, −18.0330018199674004557224348561, −17.432776936627573286043991417808, −16.96460589106339351298609430336, −16.25811221429110183346273154094, −15.3525972505781003341544287475, −14.823011171051806681771613221101, −13.85363730274913246017828661861, −13.00015313923787960898673197254, −11.7116706095735108431516254263, −11.15477675778551311347068454982, −10.611677576415048440044736223074, −9.54495829524344635571064533028, −9.090222483107877894012012619840, −7.80174551355577055919961337797, −7.12838273431480530661358048717, −6.58575427722088968329837560819, −5.38117077615358500715411200346, −4.8825043762463743626406595754, −4.09638142381942141265114420097, −2.29538091805146777486979078857, −1.251781126465073201235306442247, −0.147251211617294330213564263608, 1.31148815458410289897699605586, 1.98406155136485986933774682618, 3.07108563791608894186008764049, 4.34456928695524584712937812830, 5.10943217935824153165057135100, 6.003928139413434378517213748943, 7.259887392041049029051331191487, 7.62783428921751710954758683798, 8.754897883281291637416230434935, 9.46348451213656620986111659916, 10.54587161421467953074898798652, 10.989451283453278037778915816957, 11.743452436767419318883028148625, 12.49954502567530633351184495198, 12.91989137167434985127394767126, 14.15296388467381025302701104758, 15.20347821339776253637343393764, 16.02299121353976184837142348055, 16.84309789367789739481000595934, 17.6171052604342367551549276129, 17.89838085637361066472498198486, 18.773781135469529421247774544273, 19.370547442233403272290605865875, 20.362069273976005532450230565893, 21.07342621153782414634741875681

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