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Properties

Label 1-1375-1375.114-r0-0-0
Degree $1$
Conductor $1375$
Sign $-0.355 + 0.934i$
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  + (−0.968 + 0.248i)2-s + (−0.876 + 0.481i)3-s + (0.876 − 0.481i)4-s + (0.728 − 0.684i)6-s − 7-s + (−0.728 + 0.684i)8-s + (0.535 − 0.844i)9-s + (−0.535 + 0.844i)12-s + (−0.535 + 0.844i)13-s + (0.968 − 0.248i)14-s + (0.535 − 0.844i)16-s + (0.187 − 0.982i)17-s + (−0.309 + 0.951i)18-s + (−0.187 + 0.982i)19-s + (0.876 − 0.481i)21-s + ⋯
L(s)  = 1  + (−0.968 + 0.248i)2-s + (−0.876 + 0.481i)3-s + (0.876 − 0.481i)4-s + (0.728 − 0.684i)6-s − 7-s + (−0.728 + 0.684i)8-s + (0.535 − 0.844i)9-s + (−0.535 + 0.844i)12-s + (−0.535 + 0.844i)13-s + (0.968 − 0.248i)14-s + (0.535 − 0.844i)16-s + (0.187 − 0.982i)17-s + (−0.309 + 0.951i)18-s + (−0.187 + 0.982i)19-s + (0.876 − 0.481i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.355 + 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.355 + 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1375\)    =    \(5^{3} \cdot 11\)
Sign: $-0.355 + 0.934i$
Analytic conductor: \(6.38547\)
Root analytic conductor: \(6.38547\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1375} (114, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1375,\ (0:\ ),\ -0.355 + 0.934i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2182709357 + 0.3164144072i\)
\(L(\frac12)\) \(\approx\) \(0.2182709357 + 0.3164144072i\)
\(L(1)\) \(\approx\) \(0.4291020107 + 0.1280558384i\)
\(L(1)\) \(\approx\) \(0.4291020107 + 0.1280558384i\)

Euler product

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

−20.303144515837394708310580329091, −19.598939444038817600436040617600, −19.077022599124570208914331413673, −18.36057567275955305991841447908, −17.4938725089081109707361461526, −17.0623008633963691200218903683, −16.31572952858567384814736068671, −15.59830880019382030903050992347, −14.77424461952307453886061749141, −13.18302810754192076789855073889, −12.758391470496594468327308287576, −12.21118883704613738926448676813, −10.98619187191678068057041485938, −10.74782397981083231123545230971, −9.75554184545618431263897854454, −9.061253102928131549288610256240, −7.9227090565774914720529207251, −7.27897330926523967115037029170, −6.453783792792163039829930282602, −5.85714015259351789663154516422, −4.67934867978713871068603817150, −3.32626943066950088769736848533, −2.527440541192848180775086125986, −1.36029910702246943756891025998, −0.33718434992707486316306978975, 0.79497341363733781703774433609, 2.09413346385570754296908283161, 3.248667756446928612949331049769, 4.30237721171129625109087899087, 5.50938044394648813046936901470, 6.03381433577523606357136769293, 7.02776396842409260761628724790, 7.45539158128570412734820252730, 8.95063517122386615342841661884, 9.49930723889993895161668182396, 10.01567905382093524475636490989, 10.93346444977687122868340258117, 11.659449397776354451371898420066, 12.30537836241131021540126878275, 13.31880485502297029455160625004, 14.60384727270594060606348101783, 15.205915304056451325509035489433, 16.16825606743693723770949692474, 16.58620550885014038204112171998, 17.02986613187063125552860293498, 18.072380866845151673878253907, 18.739553075963823011442849553583, 19.27764201270625991903513485473, 20.32323183492214533404243957664, 20.9586908025074222493905947013

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