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Properties

Label 1-967-967.86-r1-0-0
Degree $1$
Conductor $967$
Sign $0.145 - 0.989i$
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  + (0.795 − 0.605i)2-s + (0.184 + 0.982i)3-s + (0.266 − 0.963i)4-s + (0.791 − 0.610i)5-s + (0.741 + 0.670i)6-s + (0.999 − 0.0325i)7-s + (−0.371 − 0.928i)8-s + (−0.932 + 0.362i)9-s + (0.260 − 0.965i)10-s + (0.560 + 0.828i)11-s + (0.996 + 0.0844i)12-s + (−0.436 − 0.899i)13-s + (0.775 − 0.631i)14-s + (0.746 + 0.665i)15-s + (−0.857 − 0.514i)16-s + (0.909 − 0.416i)17-s + ⋯
L(s)  = 1  + (0.795 − 0.605i)2-s + (0.184 + 0.982i)3-s + (0.266 − 0.963i)4-s + (0.791 − 0.610i)5-s + (0.741 + 0.670i)6-s + (0.999 − 0.0325i)7-s + (−0.371 − 0.928i)8-s + (−0.932 + 0.362i)9-s + (0.260 − 0.965i)10-s + (0.560 + 0.828i)11-s + (0.996 + 0.0844i)12-s + (−0.436 − 0.899i)13-s + (0.775 − 0.631i)14-s + (0.746 + 0.665i)15-s + (−0.857 − 0.514i)16-s + (0.909 − 0.416i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.145 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.145 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $0.145 - 0.989i$
Analytic conductor: \(103.918\)
Root analytic conductor: \(103.918\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (86, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (1:\ ),\ 0.145 - 0.989i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.731134811 - 3.221568375i\)
\(L(\frac12)\) \(\approx\) \(3.731134811 - 3.221568375i\)
\(L(1)\) \(\approx\) \(2.096833587 - 0.7360094484i\)
\(L(1)\) \(\approx\) \(2.096833587 - 0.7360094484i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (0.795 - 0.605i)T \)
3 \( 1 + (0.184 + 0.982i)T \)
5 \( 1 + (0.791 - 0.610i)T \)
7 \( 1 + (0.999 - 0.0325i)T \)
11 \( 1 + (0.560 + 0.828i)T \)
13 \( 1 + (-0.436 - 0.899i)T \)
17 \( 1 + (0.909 - 0.416i)T \)
19 \( 1 + (-0.0357 - 0.999i)T \)
23 \( 1 + (-0.763 + 0.646i)T \)
29 \( 1 + (0.864 - 0.502i)T \)
31 \( 1 + (-0.359 + 0.933i)T \)
37 \( 1 + (0.430 - 0.902i)T \)
41 \( 1 + (0.990 + 0.136i)T \)
43 \( 1 + (-0.516 - 0.856i)T \)
47 \( 1 + (-0.985 + 0.168i)T \)
53 \( 1 + (0.538 - 0.842i)T \)
59 \( 1 + (-0.322 + 0.946i)T \)
61 \( 1 + (0.613 - 0.789i)T \)
67 \( 1 + (-0.987 - 0.155i)T \)
71 \( 1 + (0.126 - 0.991i)T \)
73 \( 1 + (0.538 + 0.842i)T \)
79 \( 1 + (0.120 + 0.992i)T \)
83 \( 1 + (0.818 - 0.574i)T \)
89 \( 1 + (0.628 - 0.777i)T \)
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

−21.749880595333633863532499109163, −21.19875879332934103853052337457, −20.371219731842101883515324859555, −19.204064081318081661424593061357, −18.440578712647143395254914784166, −17.78997655940396151525957997918, −16.902817034534504477484666141056, −16.47386070529978708327683872554, −14.69536053198093676350351477696, −14.5670721159274209041489343829, −13.97467182918481785871679481594, −13.21776253977067402300129426586, −12.11945949064234813699493803827, −11.68386987503623384807835931433, −10.694142322303395319180757539515, −9.33233830983647805927289253859, −8.26701225321165784579345336118, −7.77520284889449866174646831917, −6.6919732980433168903426149386, −6.13852675023315367695162340011, −5.42287822898013164162459256705, −4.17501395113372246691760278693, −3.10594375280919046713146551919, −2.15888532202224209581389303487, −1.34402902926783316652070199236, 0.72687465648219109207470536251, 1.86992853106551291776062271669, 2.700169528763246642875348945255, 3.850435067094314263061020430437, 4.827021780081736170947593668722, 5.13018031961867939060931354398, 6.01841142001130633865826006322, 7.422163388783506434586475846656, 8.59825206857377148552812905913, 9.561626794335780562133236123349, 10.02175402860403640141378155166, 10.87389504182747224314898960127, 11.80756433348350434988393920298, 12.46874367202138689592966030925, 13.56499763298021091826236744784, 14.24756659087542733040743088060, 14.812975787477390380377171798442, 15.60895310068253802363300274895, 16.50899360668109915666642915186, 17.61145425544942691054215102130, 17.92682590557599105928206804085, 19.67871247648087089539849803878, 19.96732857904191129285967798108, 20.80899210062033170649637437667, 21.339943457464807984347244283577

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