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Properties

Label 1-967-967.17-r0-0-0
Degree $1$
Conductor $967$
Sign $0.990 + 0.138i$
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  + (0.353 − 0.935i)2-s + (0.425 + 0.905i)3-s + (−0.750 − 0.660i)4-s + (−0.995 − 0.0974i)5-s + (0.996 − 0.0779i)6-s + (−0.967 − 0.250i)7-s + (−0.883 + 0.468i)8-s + (−0.638 + 0.769i)9-s + (−0.442 + 0.896i)10-s + (0.241 − 0.970i)11-s + (0.279 − 0.960i)12-s + (0.241 + 0.970i)13-s + (−0.576 + 0.816i)14-s + (−0.334 − 0.942i)15-s + (0.126 + 0.991i)16-s + (−0.107 − 0.994i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.935i)2-s + (0.425 + 0.905i)3-s + (−0.750 − 0.660i)4-s + (−0.995 − 0.0974i)5-s + (0.996 − 0.0779i)6-s + (−0.967 − 0.250i)7-s + (−0.883 + 0.468i)8-s + (−0.638 + 0.769i)9-s + (−0.442 + 0.896i)10-s + (0.241 − 0.970i)11-s + (0.279 − 0.960i)12-s + (0.241 + 0.970i)13-s + (−0.576 + 0.816i)14-s + (−0.334 − 0.942i)15-s + (0.126 + 0.991i)16-s + (−0.107 − 0.994i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.990 + 0.138i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.990 + 0.138i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $0.990 + 0.138i$
Analytic conductor: \(4.49072\)
Root analytic conductor: \(4.49072\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (0:\ ),\ 0.990 + 0.138i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.049422351 + 0.07321047924i\)
\(L(\frac12)\) \(\approx\) \(1.049422351 + 0.07321047924i\)
\(L(1)\) \(\approx\) \(0.9276750720 - 0.1887379168i\)
\(L(1)\) \(\approx\) \(0.9276750720 - 0.1887379168i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (0.353 - 0.935i)T \)
3 \( 1 + (0.425 + 0.905i)T \)
5 \( 1 + (-0.995 - 0.0974i)T \)
7 \( 1 + (-0.967 - 0.250i)T \)
11 \( 1 + (0.241 - 0.970i)T \)
13 \( 1 + (0.241 + 0.970i)T \)
17 \( 1 + (-0.107 - 0.994i)T \)
19 \( 1 + (0.787 - 0.615i)T \)
23 \( 1 + (-0.984 + 0.174i)T \)
29 \( 1 + (-0.608 + 0.793i)T \)
31 \( 1 + (0.999 - 0.0390i)T \)
37 \( 1 + (-0.999 + 0.0195i)T \)
41 \( 1 + (0.682 + 0.730i)T \)
43 \( 1 + (0.710 + 0.703i)T \)
47 \( 1 + (-0.844 + 0.536i)T \)
53 \( 1 + (0.962 + 0.269i)T \)
59 \( 1 + (0.0487 + 0.998i)T \)
61 \( 1 + (0.682 + 0.730i)T \)
67 \( 1 + (0.999 + 0.0390i)T \)
71 \( 1 + (0.353 + 0.935i)T \)
73 \( 1 + (0.962 - 0.269i)T \)
79 \( 1 + (0.811 + 0.584i)T \)
83 \( 1 + (-0.932 - 0.362i)T \)
89 \( 1 + (0.999 + 0.0390i)T \)
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

−22.247024253288488239792786782105, −20.790803307475218724692791557084, −20.03437608758816652411075997248, −19.262719695751703130422688668648, −18.58711728703363482956552091121, −17.75955852708473794716717489642, −17.0395486193099751640973280220, −15.87554972829132187022420013611, −15.43027320117760301794953459064, −14.702370776930566669851546073957, −13.8231249037002478691773442746, −12.889673781305207315843802621297, −12.38929566249770446307791373101, −11.84246579750296554479148989007, −10.22800965553028713719585791238, −9.277548415469832124669356862272, −8.23476025895918275386687328513, −7.833872010607211275876279940220, −6.92936986096367712361140285495, −6.27885520850726812749725089086, −5.36160505463075347509936550285, −3.851703255276597763366997752266, −3.52471504514654230394771275541, −2.2758920964026520201648547284, −0.51687990601513277330576251843, 0.890763869726988668052699592905, 2.61369607701796635423634692845, 3.32578940268401310246627458862, 3.950932871508419603088004456094, 4.70971299537487376324613913315, 5.75886773444048502569750352678, 6.94811209687901608768937288920, 8.223912827092059438718727979008, 9.10509801781490947287931973589, 9.56277628685310826702919354616, 10.585423327295893904608601729267, 11.42669528685075800395812967206, 11.813758100919445416046184859184, 13.05002470515918817886002280240, 13.88045933141882601670048296092, 14.33512304599601473326579897887, 15.58631021945305243101298297237, 16.048856236818934591186989875646, 16.71425447439159618302901608153, 18.17263945328063788351180647140, 19.09675863805945447854324586221, 19.58501655978975754899731146184, 20.12120918331373382087461882588, 20.957017891454839292088870415205, 21.708124477810834724844939751537

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