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Properties

Label 1-967-967.288-r0-0-0
Degree $1$
Conductor $967$
Sign $0.834 + 0.550i$
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  + (−0.997 − 0.0714i)2-s + (0.874 + 0.485i)3-s + (0.989 + 0.142i)4-s + (−0.895 + 0.445i)5-s + (−0.837 − 0.547i)6-s + (−0.889 − 0.457i)7-s + (−0.977 − 0.212i)8-s + (0.527 + 0.849i)9-s + (0.924 − 0.380i)10-s + (−0.107 − 0.994i)11-s + (0.795 + 0.605i)12-s + (−0.807 − 0.589i)13-s + (0.854 + 0.519i)14-s + (−0.998 − 0.0455i)15-s + (0.959 + 0.282i)16-s + (−0.799 + 0.600i)17-s + ⋯
L(s)  = 1  + (−0.997 − 0.0714i)2-s + (0.874 + 0.485i)3-s + (0.989 + 0.142i)4-s + (−0.895 + 0.445i)5-s + (−0.837 − 0.547i)6-s + (−0.889 − 0.457i)7-s + (−0.977 − 0.212i)8-s + (0.527 + 0.849i)9-s + (0.924 − 0.380i)10-s + (−0.107 − 0.994i)11-s + (0.795 + 0.605i)12-s + (−0.807 − 0.589i)13-s + (0.854 + 0.519i)14-s + (−0.998 − 0.0455i)15-s + (0.959 + 0.282i)16-s + (−0.799 + 0.600i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.834 + 0.550i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.834 + 0.550i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $0.834 + 0.550i$
Analytic conductor: \(4.49072\)
Root analytic conductor: \(4.49072\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (288, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (0:\ ),\ 0.834 + 0.550i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8106707115 + 0.2431007095i\)
\(L(\frac12)\) \(\approx\) \(0.8106707115 + 0.2431007095i\)
\(L(1)\) \(\approx\) \(0.7127770035 + 0.1054251281i\)
\(L(1)\) \(\approx\) \(0.7127770035 + 0.1054251281i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (-0.997 - 0.0714i)T \)
3 \( 1 + (0.874 + 0.485i)T \)
5 \( 1 + (-0.895 + 0.445i)T \)
7 \( 1 + (-0.889 - 0.457i)T \)
11 \( 1 + (-0.107 - 0.994i)T \)
13 \( 1 + (-0.807 - 0.589i)T \)
17 \( 1 + (-0.799 + 0.600i)T \)
19 \( 1 + (0.978 + 0.206i)T \)
23 \( 1 + (0.494 - 0.869i)T \)
29 \( 1 + (0.710 + 0.703i)T \)
31 \( 1 + (-0.285 + 0.958i)T \)
37 \( 1 + (-0.395 + 0.918i)T \)
41 \( 1 + (0.203 - 0.979i)T \)
43 \( 1 + (-0.851 + 0.525i)T \)
47 \( 1 + (0.266 - 0.963i)T \)
53 \( 1 + (0.898 - 0.439i)T \)
59 \( 1 + (0.228 - 0.973i)T \)
61 \( 1 + (0.746 + 0.665i)T \)
67 \( 1 + (0.972 + 0.232i)T \)
71 \( 1 + (0.560 + 0.828i)T \)
73 \( 1 + (0.898 + 0.439i)T \)
79 \( 1 + (0.903 - 0.428i)T \)
83 \( 1 + (0.991 + 0.129i)T \)
89 \( 1 + (-0.687 + 0.726i)T \)
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

−21.3405419942127345091236725144, −20.469995252385758222325200622656, −19.73059203853589923980550513250, −19.547676316703977033627495194445, −18.645096465785679140705689327074, −17.96271239092040538304621957221, −17.00645715016887864344728334030, −16.01489619298474881744898050290, −15.44244500228741568097974992423, −14.933365700436339730010547619764, −13.66641250263837269553356629451, −12.66356865073571870661904683148, −12.05463894436400913117527343366, −11.37764096290339077443852882212, −9.82749968728321032742126748848, −9.41420916897417618013218239920, −8.78162235250872456036823655068, −7.64810212486535208897212914932, −7.27565437125139324849230685349, −6.47982291553895860704962666550, −5.031422875483645179591556641892, −3.78643067949403287425733128057, −2.75502079837890269453257831161, −2.03530646217421915592870688127, −0.67085808388744029514067965541, 0.77029445975417098512810132171, 2.43682031083294791135827496874, 3.2192835398685220159968713256, 3.70638590396212976322936024546, 5.15774907317995666943634378529, 6.69527048158296893645061651847, 7.151015820113882278723742734579, 8.254037416060297560557473575206, 8.608725570869949921248618106281, 9.72952709240528482428119732371, 10.44495179637138858983375190587, 10.93242808984267219394534216995, 12.100374575771623028509549745104, 12.967095806498377134155413090466, 14.06902725951861873993919439842, 14.93696818834938414577924230791, 15.68556370887798154981276917586, 16.20762813816453963360092473533, 16.90109346395106088061585110420, 18.13514163972744664901524371493, 18.92559783325792130221895326441, 19.55857749225549951227657226175, 19.93917166203074505120144025483, 20.65122423313576361089763170851, 21.849854786174159329857268476137

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