Properties

Label 1-919-919.512-r0-0-0
Degree $1$
Conductor $919$
Sign $-0.962 + 0.271i$
Analytic cond. $4.26781$
Root an. cond. $4.26781$
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  + (0.445 − 0.895i)2-s + (−0.982 + 0.183i)3-s + (−0.602 − 0.798i)4-s + (−0.982 + 0.183i)5-s + (−0.273 + 0.961i)6-s + (−0.273 + 0.961i)7-s + (−0.982 + 0.183i)8-s + (0.932 − 0.361i)9-s + (−0.273 + 0.961i)10-s + (0.932 + 0.361i)11-s + (0.739 + 0.673i)12-s + (−0.602 + 0.798i)13-s + (0.739 + 0.673i)14-s + (0.932 − 0.361i)15-s + (−0.273 + 0.961i)16-s + (0.0922 − 0.995i)17-s + ⋯
L(s)  = 1  + (0.445 − 0.895i)2-s + (−0.982 + 0.183i)3-s + (−0.602 − 0.798i)4-s + (−0.982 + 0.183i)5-s + (−0.273 + 0.961i)6-s + (−0.273 + 0.961i)7-s + (−0.982 + 0.183i)8-s + (0.932 − 0.361i)9-s + (−0.273 + 0.961i)10-s + (0.932 + 0.361i)11-s + (0.739 + 0.673i)12-s + (−0.602 + 0.798i)13-s + (0.739 + 0.673i)14-s + (0.932 − 0.361i)15-s + (−0.273 + 0.961i)16-s + (0.0922 − 0.995i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 919 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.962 + 0.271i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 919 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.962 + 0.271i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(919\)
Sign: $-0.962 + 0.271i$
Analytic conductor: \(4.26781\)
Root analytic conductor: \(4.26781\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{919} (512, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 919,\ (0:\ ),\ -0.962 + 0.271i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01799065741 - 0.1298581036i\)
\(L(\frac12)\) \(\approx\) \(0.01799065741 - 0.1298581036i\)
\(L(1)\) \(\approx\) \(0.5887341663 - 0.1981191723i\)
\(L(1)\) \(\approx\) \(0.5887341663 - 0.1981191723i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad919 \( 1 \)
good2 \( 1 + (0.445 - 0.895i)T \)
3 \( 1 + (-0.982 + 0.183i)T \)
5 \( 1 + (-0.982 + 0.183i)T \)
7 \( 1 + (-0.273 + 0.961i)T \)
11 \( 1 + (0.932 + 0.361i)T \)
13 \( 1 + (-0.602 + 0.798i)T \)
17 \( 1 + (0.0922 - 0.995i)T \)
19 \( 1 + (-0.982 - 0.183i)T \)
23 \( 1 + (0.932 + 0.361i)T \)
29 \( 1 + (0.445 - 0.895i)T \)
31 \( 1 + (-0.273 + 0.961i)T \)
37 \( 1 + (-0.602 + 0.798i)T \)
41 \( 1 + (0.739 - 0.673i)T \)
43 \( 1 + (-0.982 - 0.183i)T \)
47 \( 1 + (-0.602 - 0.798i)T \)
53 \( 1 + T \)
59 \( 1 + (-0.602 - 0.798i)T \)
61 \( 1 + (-0.273 - 0.961i)T \)
67 \( 1 + (-0.850 + 0.526i)T \)
71 \( 1 + (-0.273 + 0.961i)T \)
73 \( 1 + (-0.850 + 0.526i)T \)
79 \( 1 + (-0.273 - 0.961i)T \)
83 \( 1 + (0.0922 + 0.995i)T \)
89 \( 1 + (-0.982 - 0.183i)T \)
97 \( 1 + (0.932 - 0.361i)T \)
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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

−22.825577088185378074842163790408, −21.85025994943206494240702887053, −20.991616390672055995733370204320, −19.71337015860090640483648052309, −19.22767415784498168241868973125, −18.07263869764711653292748376918, −17.1584335517864566776478140975, −16.74942656730501251046343170692, −16.217866237768174894591335701, −15.0810943227742087618283775398, −14.64758035724549380286599026193, −13.30448525953917132299008897397, −12.71739225868510380581094080536, −12.10062959403223822634529221504, −11.072024859892834544333834891989, −10.34559367202745730363183944888, −9.02472349031577235071004478266, −8.02807725255827509043196390648, −7.29819449040639319850144152018, −6.61080973210137416706943933906, −5.81007131926972349237620227193, −4.648693077832644116245038692842, −4.14077106903952610665604614464, −3.22893798058884013183336750464, −1.10994806623897185849228849972, 0.068270425630645068344951090811, 1.51621176577156589965808870940, 2.71507189645326446455983189635, 3.78032306600796991394034377738, 4.61636812975368862317266508177, 5.23282795607628358825856435573, 6.486340195749313773979617551730, 7.009767916613570209923740827881, 8.68163275408108170935138914724, 9.38544664222427918819782138685, 10.25120551513836236803451718941, 11.29402590638047107029921831175, 11.82831947146922101200819719909, 12.17627183420181043687066263694, 13.05902470556543910133743427636, 14.33897347612433874842760723676, 15.11259310201361190999116686188, 15.69297593550538670010726380230, 16.72860905425510724260112099939, 17.611820445775760010150016507, 18.60010655510634369337427866062, 19.11517350890786205952735385713, 19.73818359074371641080546969105, 20.86781284112145674748063345230, 21.67344378510751685600454120548

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