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Properties

Label 1-919-919.284-r0-0-0
Degree $1$
Conductor $919$
Sign $0.837 + 0.546i$
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  + (0.739 − 0.673i)2-s + (−0.602 − 0.798i)3-s + (0.0922 − 0.995i)4-s + (−0.602 − 0.798i)5-s + (−0.982 − 0.183i)6-s + (−0.982 − 0.183i)7-s + (−0.602 − 0.798i)8-s + (−0.273 + 0.961i)9-s + (−0.982 − 0.183i)10-s + (−0.273 − 0.961i)11-s + (−0.850 + 0.526i)12-s + (0.0922 + 0.995i)13-s + (−0.850 + 0.526i)14-s + (−0.273 + 0.961i)15-s + (−0.982 − 0.183i)16-s + (0.445 + 0.895i)17-s + ⋯
L(s)  = 1  + (0.739 − 0.673i)2-s + (−0.602 − 0.798i)3-s + (0.0922 − 0.995i)4-s + (−0.602 − 0.798i)5-s + (−0.982 − 0.183i)6-s + (−0.982 − 0.183i)7-s + (−0.602 − 0.798i)8-s + (−0.273 + 0.961i)9-s + (−0.982 − 0.183i)10-s + (−0.273 − 0.961i)11-s + (−0.850 + 0.526i)12-s + (0.0922 + 0.995i)13-s + (−0.850 + 0.526i)14-s + (−0.273 + 0.961i)15-s + (−0.982 − 0.183i)16-s + (0.445 + 0.895i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 919 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.837 + 0.546i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 919 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.837 + 0.546i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(919\)
Sign: $0.837 + 0.546i$
Analytic conductor: \(4.26781\)
Root analytic conductor: \(4.26781\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{919} (284, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 919,\ (0:\ ),\ 0.837 + 0.546i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.001601027671 + 0.0004765820772i\)
\(L(\frac12)\) \(\approx\) \(0.001601027671 + 0.0004765820772i\)
\(L(1)\) \(\approx\) \(0.5205594258 - 0.5800632127i\)
\(L(1)\) \(\approx\) \(0.5205594258 - 0.5800632127i\)

Euler product

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

−22.08193981705864448575819198968, −21.423586534612766643946315761607, −20.29845653431048326479926592985, −19.73191354511710620406394949770, −18.24101483400099762417567452403, −17.83313856803135405895199218131, −16.77556686979202425416161533244, −15.96160004779234394352956638399, −15.43252498302058426195414112711, −15.01260310727296203862589126567, −13.996798058233073312745378304412, −12.85282093786553708015268762985, −12.25623351403226142976935003702, −11.4151808047898705384384312919, −10.50695219343791535482560615551, −9.6947044093681224833286485405, −8.659101958936763223406496755879, −7.35128569329365873741390810523, −6.919473717502583591620954483880, −5.880770007190537580076590443, −5.155971812087595668982119088628, −4.16647317319740774966513044566, −3.31309515401994228173520394627, −2.7026888251766093040058664360, −0.00067014020375354999814887573, 1.11888205274887335295247546251, 2.141173519510793663393027380945, 3.434779706261632924688351166718, 4.18960730556553685240409439414, 5.24737603998244011223657483752, 6.159328676021600704643396479590, 6.65872844939870324868587075227, 8.01660482090792375858759309872, 8.82743811607081788065791815773, 10.0851823162618700654571422876, 10.811868073955387788740160133631, 11.82113829526230421679165761988, 12.23769228036502127251278513235, 13.06600505449355795229703835399, 13.53039809327209270318396952236, 14.52111413090835532251848788541, 15.68914125393777310178976578325, 16.60148274872308377930342678733, 16.80192172103177718906630163632, 18.53849414003282834913150397808, 18.91846926227878070478137045590, 19.53420364720486295748481344716, 20.28129973405728460463515147068, 21.336717794600197135808364555571, 21.92583271560522737319810624960

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