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Properties

Label 1-919-919.849-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: $\chi_{919} (849, \cdot )$
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

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

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