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Properties

Label 1-967-967.446-r0-0-0
Degree $1$
Conductor $967$
Sign $0.991 - 0.132i$
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  + (−0.107 − 0.994i)2-s + (−0.724 + 0.689i)3-s + (−0.977 + 0.212i)4-s + (−0.638 + 0.769i)5-s + (0.763 + 0.646i)6-s + (0.653 + 0.756i)7-s + (0.316 + 0.948i)8-s + (0.0487 − 0.998i)9-s + (0.833 + 0.552i)10-s + (0.811 − 0.584i)11-s + (0.560 − 0.828i)12-s + (0.811 + 0.584i)13-s + (0.682 − 0.730i)14-s + (−0.0682 − 0.997i)15-s + (0.909 − 0.416i)16-s + (−0.822 + 0.568i)17-s + ⋯
L(s)  = 1  + (−0.107 − 0.994i)2-s + (−0.724 + 0.689i)3-s + (−0.977 + 0.212i)4-s + (−0.638 + 0.769i)5-s + (0.763 + 0.646i)6-s + (0.653 + 0.756i)7-s + (0.316 + 0.948i)8-s + (0.0487 − 0.998i)9-s + (0.833 + 0.552i)10-s + (0.811 − 0.584i)11-s + (0.560 − 0.828i)12-s + (0.811 + 0.584i)13-s + (0.682 − 0.730i)14-s + (−0.0682 − 0.997i)15-s + (0.909 − 0.416i)16-s + (−0.822 + 0.568i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.132i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.132i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $0.991 - 0.132i$
Analytic conductor: \(4.49072\)
Root analytic conductor: \(4.49072\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (446, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (0:\ ),\ 0.991 - 0.132i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.004425389 - 0.06674210349i\)
\(L(\frac12)\) \(\approx\) \(1.004425389 - 0.06674210349i\)
\(L(1)\) \(\approx\) \(0.7961752471 - 0.07505479618i\)
\(L(1)\) \(\approx\) \(0.7961752471 - 0.07505479618i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (-0.107 - 0.994i)T \)
3 \( 1 + (-0.724 + 0.689i)T \)
5 \( 1 + (-0.638 + 0.769i)T \)
7 \( 1 + (0.653 + 0.756i)T \)
11 \( 1 + (0.811 - 0.584i)T \)
13 \( 1 + (0.811 + 0.584i)T \)
17 \( 1 + (-0.822 + 0.568i)T \)
19 \( 1 + (0.951 - 0.307i)T \)
23 \( 1 + (0.00975 - 0.999i)T \)
29 \( 1 + (0.389 - 0.921i)T \)
31 \( 1 + (0.938 + 0.344i)T \)
37 \( 1 + (-0.984 - 0.174i)T \)
41 \( 1 + (0.460 - 0.887i)T \)
43 \( 1 + (0.737 - 0.675i)T \)
47 \( 1 + (-0.371 + 0.928i)T \)
53 \( 1 + (-0.775 - 0.631i)T \)
59 \( 1 + (0.425 - 0.905i)T \)
61 \( 1 + (0.460 - 0.887i)T \)
67 \( 1 + (0.938 - 0.344i)T \)
71 \( 1 + (-0.107 + 0.994i)T \)
73 \( 1 + (-0.775 + 0.631i)T \)
79 \( 1 + (0.787 + 0.615i)T \)
83 \( 1 + (0.981 - 0.193i)T \)
89 \( 1 + (0.938 - 0.344i)T \)
97 \( 1 + (-0.222 + 0.974i)T \)
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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

−22.23847529446202047236941182013, −20.91061615393919544154083001896, −19.9691657205897776851276800450, −19.40120195442673083351486837669, −18.18825379834488580370026221617, −17.7069603916665762526171612056, −17.12373730083313012182421473693, −16.24048722095800402284066593982, −15.75415036551060876551275855981, −14.67684304016448480182373819528, −13.635427508866246175913176572235, −13.26264504224884328876930862545, −12.152212617328879748387990226980, −11.5001289645680201512618594871, −10.50623456418243992822268140890, −9.36764166535646979172966292075, −8.41735065287710604542457037068, −7.66390883749038933007001908051, −7.12520034766953007195881961202, −6.18297229402625452020514117772, −5.129751246416630983745432287673, −4.60777465988090642788296756195, −3.62098060843868773415105600182, −1.419330918206406279491188788142, −0.88743045793397504736619162541, 0.801639252583922959035545605277, 2.1490437802609797193884104111, 3.29116049450281487843941727915, 4.044382531314917563110877966776, 4.76569555882724088143358278827, 5.922469309164193235810852184513, 6.730151439339526604379563464771, 8.27047235802696954629023997269, 8.82542723164693286068128177199, 9.749051137253639494334255067897, 10.853004987970374465119185929067, 11.184115345623380728820186164336, 11.81391896390576585019913815735, 12.46768376409043224617222730385, 13.94290116580039336098358071369, 14.43019274235703458599280428908, 15.52472639414742816324058502443, 16.07763879476825022277503922146, 17.42240077252719621147105994896, 17.73815026735471982806506913342, 18.8386211113111731133801046458, 19.13549340243682738629069251123, 20.35712131405817671696457713277, 21.023404971463966615521383135756, 21.825274795962689850671755518312

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