Properties

Label 1-967-967.129-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: $\chi_{967} (129, \cdot )$
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

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

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