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Properties

Label 1-967-967.456-r0-0-0
Degree $1$
Conductor $967$
Sign $0.0322 + 0.999i$
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  + (−0.767 + 0.641i)2-s + (0.592 − 0.805i)3-s + (0.177 − 0.984i)4-s + (0.978 + 0.206i)5-s + (0.0617 + 0.998i)6-s + (0.190 + 0.981i)7-s + (0.494 + 0.869i)8-s + (−0.297 − 0.954i)9-s + (−0.883 + 0.468i)10-s + (−0.864 + 0.502i)11-s + (−0.687 − 0.726i)12-s + (−0.00325 + 0.999i)13-s + (−0.775 − 0.631i)14-s + (0.746 − 0.665i)15-s + (−0.936 − 0.350i)16-s + (−0.999 + 0.0195i)17-s + ⋯
L(s)  = 1  + (−0.767 + 0.641i)2-s + (0.592 − 0.805i)3-s + (0.177 − 0.984i)4-s + (0.978 + 0.206i)5-s + (0.0617 + 0.998i)6-s + (0.190 + 0.981i)7-s + (0.494 + 0.869i)8-s + (−0.297 − 0.954i)9-s + (−0.883 + 0.468i)10-s + (−0.864 + 0.502i)11-s + (−0.687 − 0.726i)12-s + (−0.00325 + 0.999i)13-s + (−0.775 − 0.631i)14-s + (0.746 − 0.665i)15-s + (−0.936 − 0.350i)16-s + (−0.999 + 0.0195i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0322 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0322 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $0.0322 + 0.999i$
Analytic conductor: \(4.49072\)
Root analytic conductor: \(4.49072\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (456, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (0:\ ),\ 0.0322 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8767922840 + 0.8489845248i\)
\(L(\frac12)\) \(\approx\) \(0.8767922840 + 0.8489845248i\)
\(L(1)\) \(\approx\) \(0.9288883709 + 0.2798108434i\)
\(L(1)\) \(\approx\) \(0.9288883709 + 0.2798108434i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (-0.767 + 0.641i)T \)
3 \( 1 + (0.592 - 0.805i)T \)
5 \( 1 + (0.978 + 0.206i)T \)
7 \( 1 + (0.190 + 0.981i)T \)
11 \( 1 + (-0.864 + 0.502i)T \)
13 \( 1 + (-0.00325 + 0.999i)T \)
17 \( 1 + (-0.999 + 0.0195i)T \)
19 \( 1 + (0.966 + 0.257i)T \)
23 \( 1 + (-0.967 - 0.250i)T \)
29 \( 1 + (0.560 + 0.828i)T \)
31 \( 1 + (0.728 + 0.684i)T \)
37 \( 1 + (-0.783 + 0.620i)T \)
41 \( 1 + (-0.990 + 0.136i)T \)
43 \( 1 + (-0.0941 + 0.995i)T \)
47 \( 1 + (-0.0552 - 0.998i)T \)
53 \( 1 + (0.538 + 0.842i)T \)
59 \( 1 + (0.994 - 0.103i)T \)
61 \( 1 + (0.613 + 0.789i)T \)
67 \( 1 + (-0.957 - 0.288i)T \)
71 \( 1 + (0.938 - 0.344i)T \)
73 \( 1 + (0.538 - 0.842i)T \)
79 \( 1 + (-0.851 + 0.525i)T \)
83 \( 1 + (-0.986 - 0.161i)T \)
89 \( 1 + (0.228 + 0.973i)T \)
97 \( 1 + (-0.900 + 0.433i)T \)
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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

−21.22300747491557660060098033018, −20.65833104739365672072435995835, −20.19270307177358616479012942704, −19.45506610324549664500609805963, −18.33690574169168773206954956614, −17.561869341675711114939474346097, −17.08901800673412609932070893088, −15.98768530191171562357822840179, −15.61054058569295906319976373689, −14.11242224735388748099454196239, −13.50768113005163717830697074653, −13.03490808316527255063057774550, −11.58685735831793550101034881667, −10.706801322131782182466207891655, −10.138638439737163961217333742057, −9.688130187990327288634948087822, −8.55203319040689362595161977883, −8.057865343282684324982019738095, −7.05453396820394947768678257263, −5.65339149404092781749497895019, −4.68714954052452770782722623094, −3.684854242183130120840124480953, −2.77235574063807076864078623912, −1.990295943047820267570223533909, −0.59859679957205471389008662391, 1.47865983085566486543836868958, 2.08938172675227554224925220329, 2.86047353538880428228919576870, 4.801212829014505519454100041860, 5.65729566277956766732041138270, 6.56519586705928926499058856653, 7.0589924161032872588241207144, 8.25388779786764122754876228622, 8.76131041679302241681237603414, 9.59196507430928060530104266657, 10.28535476343379538075897788670, 11.53634985830414776684610839920, 12.35028171497919289851201079671, 13.53695846485863877440072382204, 13.99064787079808499011983621717, 14.84572701552965303552314654947, 15.55746969424050761214617849495, 16.45474191854541964517911062279, 17.62978685556549367605769807783, 18.12801735115279092593901030943, 18.428205196393687172415573507329, 19.34962542845170939974502125739, 20.21972024610488573537837887619, 20.97540255695860678060210170234, 21.89096301935074569021323134331

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