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Properties

Label 1-919-919.229-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: $\chi_{919} (229, \cdot )$
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

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

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