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Properties

Label 1-967-967.15-r0-0-0
Degree $1$
Conductor $967$
Sign $0.957 + 0.287i$
Analytic cond. $4.49072$
Root an. cond. $4.49072$
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  + (0.613 + 0.789i)2-s + (−0.775 + 0.631i)3-s + (−0.247 + 0.968i)4-s + (0.0227 + 0.999i)5-s + (−0.974 − 0.225i)6-s + (−0.998 − 0.0455i)7-s + (−0.917 + 0.398i)8-s + (0.203 − 0.979i)9-s + (−0.775 + 0.631i)10-s + (0.203 − 0.979i)11-s + (−0.419 − 0.907i)12-s + (−0.949 − 0.313i)13-s + (−0.576 − 0.816i)14-s + (−0.648 − 0.761i)15-s + (−0.877 − 0.480i)16-s + (−0.334 − 0.942i)17-s + ⋯
L(s)  = 1  + (0.613 + 0.789i)2-s + (−0.775 + 0.631i)3-s + (−0.247 + 0.968i)4-s + (0.0227 + 0.999i)5-s + (−0.974 − 0.225i)6-s + (−0.998 − 0.0455i)7-s + (−0.917 + 0.398i)8-s + (0.203 − 0.979i)9-s + (−0.775 + 0.631i)10-s + (0.203 − 0.979i)11-s + (−0.419 − 0.907i)12-s + (−0.949 − 0.313i)13-s + (−0.576 − 0.816i)14-s + (−0.648 − 0.761i)15-s + (−0.877 − 0.480i)16-s + (−0.334 − 0.942i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.957 + 0.287i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.957 + 0.287i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $0.957 + 0.287i$
Analytic conductor: \(4.49072\)
Root analytic conductor: \(4.49072\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (15, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (0:\ ),\ 0.957 + 0.287i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6749817411 + 0.09926390457i\)
\(L(\frac12)\) \(\approx\) \(0.6749817411 + 0.09926390457i\)
\(L(1)\) \(\approx\) \(0.6701125716 + 0.4901770941i\)
\(L(1)\) \(\approx\) \(0.6701125716 + 0.4901770941i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (0.613 + 0.789i)T \)
3 \( 1 + (-0.775 + 0.631i)T \)
5 \( 1 + (0.0227 + 0.999i)T \)
7 \( 1 + (-0.998 - 0.0455i)T \)
11 \( 1 + (0.203 - 0.979i)T \)
13 \( 1 + (-0.949 - 0.313i)T \)
17 \( 1 + (-0.334 - 0.942i)T \)
19 \( 1 + (0.934 + 0.356i)T \)
23 \( 1 + (0.962 - 0.269i)T \)
29 \( 1 + (0.203 + 0.979i)T \)
31 \( 1 + (0.113 - 0.993i)T \)
37 \( 1 + (-0.949 - 0.313i)T \)
41 \( 1 + (0.682 - 0.730i)T \)
43 \( 1 + (0.983 - 0.181i)T \)
47 \( 1 + (-0.648 - 0.761i)T \)
53 \( 1 + (-0.715 - 0.699i)T \)
59 \( 1 + (-0.715 + 0.699i)T \)
61 \( 1 + (0.291 + 0.956i)T \)
67 \( 1 + (-0.917 - 0.398i)T \)
71 \( 1 + (-0.990 - 0.136i)T \)
73 \( 1 + (-0.715 + 0.699i)T \)
79 \( 1 + (-0.715 + 0.699i)T \)
83 \( 1 + (-0.974 - 0.225i)T \)
89 \( 1 + (0.803 - 0.595i)T \)
97 \( 1 + T \)
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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

−21.85890581606005932470541657200, −20.999328649157675309220198904302, −19.96247071818377377558604457319, −19.49550787692213670668325900197, −18.9173927608056219065994714218, −17.56280425458478951423141343542, −17.31787189488907693890458192060, −16.111807281127565788985800036699, −15.475147273176941666736389223407, −14.285113285531806656112065087411, −13.28906267843723033358514411015, −12.79107438400814008733887539321, −12.23768395891506028398141154264, −11.63410368827234095196976872016, −10.50328808494818027752555740398, −9.68252690021880257728389594357, −9.06182481867915164329467755099, −7.59378181036705422904675625662, −6.65516264380169019220744753749, −5.87086749467364143702156368242, −4.88799377216216104777182965474, −4.41465232084294484499498403743, −3.01824919977894876491648023930, −1.92112057680695433311021229857, −1.071486983641055890150157903839, 0.30043652016104098840382438438, 2.88081547537163408648548968473, 3.260239612284701102165518713038, 4.305314134055673753648847805038, 5.43423895863888488282228850012, 5.94737774074632705648068794100, 6.96135933511845516161398555765, 7.30281138409118807409955863334, 8.88484768396499150848188352330, 9.62355341397247146251337671542, 10.54444014279504104181191849917, 11.45919295187952015850806092678, 12.146584874394903071952770214815, 13.105826017461892948463947158984, 14.0357931514101156543595247447, 14.709449928937074673965040083091, 15.61002174923847647185014738036, 16.13027533042245196408575886332, 16.82804346572654132356998996301, 17.65001418010896254696860134566, 18.42865246670289616198307644887, 19.28471252466852905442330811461, 20.489593680503430085997930180705, 21.42181222448011999824884603865, 22.15428106354429716629946405157

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