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Properties

Label 1-967-967.89-r1-0-0
Degree $1$
Conductor $967$
Sign $0.233 + 0.972i$
Analytic cond. $103.918$
Root an. cond. $103.918$
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  + (−0.941 − 0.337i)2-s + (0.297 − 0.954i)3-s + (0.771 + 0.636i)4-s + (0.807 + 0.589i)5-s + (−0.602 + 0.797i)6-s + (−0.139 − 0.990i)7-s + (−0.511 − 0.859i)8-s + (−0.822 − 0.568i)9-s + (−0.560 − 0.828i)10-s + (0.494 + 0.869i)11-s + (0.837 − 0.547i)12-s + (−0.505 − 0.862i)13-s + (−0.203 + 0.979i)14-s + (0.803 − 0.595i)15-s + (0.190 + 0.981i)16-s + (0.999 + 0.0390i)17-s + ⋯
L(s)  = 1  + (−0.941 − 0.337i)2-s + (0.297 − 0.954i)3-s + (0.771 + 0.636i)4-s + (0.807 + 0.589i)5-s + (−0.602 + 0.797i)6-s + (−0.139 − 0.990i)7-s + (−0.511 − 0.859i)8-s + (−0.822 − 0.568i)9-s + (−0.560 − 0.828i)10-s + (0.494 + 0.869i)11-s + (0.837 − 0.547i)12-s + (−0.505 − 0.862i)13-s + (−0.203 + 0.979i)14-s + (0.803 − 0.595i)15-s + (0.190 + 0.981i)16-s + (0.999 + 0.0390i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.233 + 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.233 + 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $0.233 + 0.972i$
Analytic conductor: \(103.918\)
Root analytic conductor: \(103.918\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (89, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (1:\ ),\ 0.233 + 0.972i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3309272663 + 0.2608141822i\)
\(L(\frac12)\) \(\approx\) \(0.3309272663 + 0.2608141822i\)
\(L(1)\) \(\approx\) \(0.6985124542 - 0.2746813586i\)
\(L(1)\) \(\approx\) \(0.6985124542 - 0.2746813586i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (-0.941 - 0.337i)T \)
3 \( 1 + (0.297 - 0.954i)T \)
5 \( 1 + (0.807 + 0.589i)T \)
7 \( 1 + (-0.139 - 0.990i)T \)
11 \( 1 + (0.494 + 0.869i)T \)
13 \( 1 + (-0.505 - 0.862i)T \)
17 \( 1 + (0.999 + 0.0390i)T \)
19 \( 1 + (0.00325 - 0.999i)T \)
23 \( 1 + (-0.874 + 0.485i)T \)
29 \( 1 + (0.371 + 0.928i)T \)
31 \( 1 + (-0.895 + 0.445i)T \)
37 \( 1 + (-0.728 + 0.684i)T \)
41 \( 1 + (-0.962 - 0.269i)T \)
43 \( 1 + (-0.328 + 0.944i)T \)
47 \( 1 + (-0.401 - 0.915i)T \)
53 \( 1 + (0.995 + 0.0909i)T \)
59 \( 1 + (-0.310 - 0.950i)T \)
61 \( 1 + (-0.715 + 0.699i)T \)
67 \( 1 + (-0.833 + 0.552i)T \)
71 \( 1 + (0.763 + 0.646i)T \)
73 \( 1 + (0.995 - 0.0909i)T \)
79 \( 1 + (-0.549 + 0.835i)T \)
83 \( 1 + (-0.197 + 0.980i)T \)
89 \( 1 + (-0.0617 - 0.998i)T \)
97 \( 1 + (0.623 + 0.781i)T \)
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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

−21.310496783392124574313269447788, −20.67619556559286507277871447621, −19.73650138317930507497462973461, −18.97232982382712578490751134822, −18.31728816008554240377316010055, −17.15844724808658621757979250641, −16.55257764191597215139222005209, −16.22592918919349913288729951833, −15.16830336340719047896636333153, −14.35632884045998728523190079481, −13.86836190146493016095844888762, −12.220686686850349103100504609610, −11.71016311898472065500100863817, −10.48815526896793635041109047471, −9.81568107240355606429679923814, −9.18576362948516166445774790164, −8.623665081303264022875203234343, −7.83600907242756601290081635393, −6.25779145638210180618954352736, −5.78953547538441605376471305040, −4.97172923386294856245877547078, −3.61804744250981962490786525158, −2.419067831963918265184268284297, −1.66294765161974414206698533752, −0.112259048753809936504283398375, 1.14474693048563463691787798208, 1.811477457375765600798153214861, 2.885052482517013063199117368463, 3.5701470845259586228373593194, 5.322269902877098155524274231200, 6.56888501230470801463117980295, 7.0730552660158487406917700258, 7.66321580885952851962777295056, 8.70057174915782982403324655739, 9.774264942937822308498276057232, 10.158477373925316485911877880040, 11.14554294777005714144994410641, 12.13326102829172967958793691601, 12.83165977896205452946038067807, 13.663028755767146319330221481365, 14.49945959452699520505251887746, 15.29678489500370791644536550095, 16.7085106417984025119347744925, 17.260769455166072208784404129322, 17.93097460264110000779100588814, 18.35263713718050073776950638634, 19.53963844154158215760274046459, 19.8976021685595236363388533090, 20.54613955060374912681085550815, 21.6043077982214203953129271660

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