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Properties

Label 1-919-919.305-r0-0-0
Degree $1$
Conductor $919$
Sign $-0.558 + 0.829i$
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  + (−0.602 + 0.798i)2-s + (0.932 + 0.361i)3-s + (−0.273 − 0.961i)4-s + (0.932 + 0.361i)5-s + (−0.850 + 0.526i)6-s + (−0.850 + 0.526i)7-s + (0.932 + 0.361i)8-s + (0.739 + 0.673i)9-s + (−0.850 + 0.526i)10-s + (0.739 − 0.673i)11-s + (0.0922 − 0.995i)12-s + (−0.273 + 0.961i)13-s + (0.0922 − 0.995i)14-s + (0.739 + 0.673i)15-s + (−0.850 + 0.526i)16-s + (−0.982 + 0.183i)17-s + ⋯
L(s)  = 1  + (−0.602 + 0.798i)2-s + (0.932 + 0.361i)3-s + (−0.273 − 0.961i)4-s + (0.932 + 0.361i)5-s + (−0.850 + 0.526i)6-s + (−0.850 + 0.526i)7-s + (0.932 + 0.361i)8-s + (0.739 + 0.673i)9-s + (−0.850 + 0.526i)10-s + (0.739 − 0.673i)11-s + (0.0922 − 0.995i)12-s + (−0.273 + 0.961i)13-s + (0.0922 − 0.995i)14-s + (0.739 + 0.673i)15-s + (−0.850 + 0.526i)16-s + (−0.982 + 0.183i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 919 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.558 + 0.829i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 919 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.558 + 0.829i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(919\)
Sign: $-0.558 + 0.829i$
Analytic conductor: \(4.26781\)
Root analytic conductor: \(4.26781\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{919} (305, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 919,\ (0:\ ),\ -0.558 + 0.829i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7663434869 + 1.440604663i\)
\(L(\frac12)\) \(\approx\) \(0.7663434869 + 1.440604663i\)
\(L(1)\) \(\approx\) \(0.9507038660 + 0.7045014541i\)
\(L(1)\) \(\approx\) \(0.9507038660 + 0.7045014541i\)

Euler product

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

−21.25389067948311795054711501649, −20.600541380249736524665159510045, −19.92693383844434671516424063751, −19.58031753783047991238605325975, −18.51380001433943653719360384823, −17.72019724448431621150894231496, −17.22716589578350189503825843563, −16.221385122525164327454689376148, −15.2187400731130353787123294392, −14.05652319040416774144666223094, −13.39085677370631150574413526973, −12.83451075414604978165236517052, −12.17761894777431428652070413514, −10.849982243235294954887720066082, −9.92136168900390641433506508849, −9.3598240565433432882346929635, −8.94710225893149688738576783050, −7.554742766777565549284632210547, −7.16019460143154658995185157707, −5.89280391341947979195811908240, −4.42779710303622587824893421354, −3.54276280257257638547803038157, −2.64363432177104516225323450017, −1.795940460956807145423709876937, −0.81177320964015609704581963177, 1.454178456679746055080968054539, 2.41553716369808311222294638042, 3.42798084075221340647408782405, 4.70468981848780444733316414228, 5.67765669841068255904750064878, 6.73866271577904044378856849081, 7.04228594724404491028795347527, 8.61285319271300374812950328780, 9.074913746512360415821152514, 9.543011924638678985107728868708, 10.43805144844667373187207036465, 11.362739842939609850644554000293, 12.96183555121017196064729017566, 13.619439798968041942363988269301, 14.37084836785414532578281719479, 14.90666155756389423664348631881, 15.90162134498863558178935657034, 16.49715095639843583428177495195, 17.263129540288030575284410269470, 18.50289945667782029180279414777, 18.74444103782414688032299369821, 19.6958903314372239394713541942, 20.2880078957531960645890675376, 21.739345218525819946541463526684, 21.90176794723318400872357254245

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