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Properties

Label 1-967-967.183-r1-0-0
Degree $1$
Conductor $967$
Sign $0.560 - 0.828i$
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  + (0.886 + 0.462i)2-s + (−0.833 + 0.552i)3-s + (0.571 + 0.820i)4-s + (0.383 + 0.923i)5-s + (−0.994 + 0.103i)6-s + (−0.943 − 0.331i)7-s + (0.126 + 0.991i)8-s + (0.389 − 0.921i)9-s + (−0.0876 + 0.996i)10-s + (−0.750 + 0.660i)11-s + (−0.929 − 0.368i)12-s + (−0.947 + 0.319i)13-s + (−0.682 − 0.730i)14-s + (−0.829 − 0.557i)15-s + (−0.347 + 0.937i)16-s + (−0.371 + 0.928i)17-s + ⋯
L(s)  = 1  + (0.886 + 0.462i)2-s + (−0.833 + 0.552i)3-s + (0.571 + 0.820i)4-s + (0.383 + 0.923i)5-s + (−0.994 + 0.103i)6-s + (−0.943 − 0.331i)7-s + (0.126 + 0.991i)8-s + (0.389 − 0.921i)9-s + (−0.0876 + 0.996i)10-s + (−0.750 + 0.660i)11-s + (−0.929 − 0.368i)12-s + (−0.947 + 0.319i)13-s + (−0.682 − 0.730i)14-s + (−0.829 − 0.557i)15-s + (−0.347 + 0.937i)16-s + (−0.371 + 0.928i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.560 - 0.828i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.560 - 0.828i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $0.560 - 0.828i$
Analytic conductor: \(103.918\)
Root analytic conductor: \(103.918\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (183, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (1:\ ),\ 0.560 - 0.828i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.6930680751 + 0.3678015548i\)
\(L(\frac12)\) \(\approx\) \(-0.6930680751 + 0.3678015548i\)
\(L(1)\) \(\approx\) \(0.6453436624 + 0.7927126838i\)
\(L(1)\) \(\approx\) \(0.6453436624 + 0.7927126838i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (0.886 + 0.462i)T \)
3 \( 1 + (-0.833 + 0.552i)T \)
5 \( 1 + (0.383 + 0.923i)T \)
7 \( 1 + (-0.943 - 0.331i)T \)
11 \( 1 + (-0.750 + 0.660i)T \)
13 \( 1 + (-0.947 + 0.319i)T \)
17 \( 1 + (-0.371 + 0.928i)T \)
19 \( 1 + (-0.633 + 0.773i)T \)
23 \( 1 + (-0.972 + 0.232i)T \)
29 \( 1 + (0.984 - 0.174i)T \)
31 \( 1 + (0.998 - 0.0520i)T \)
37 \( 1 + (0.522 + 0.852i)T \)
41 \( 1 + (-0.460 - 0.887i)T \)
43 \( 1 + (0.999 - 0.00650i)T \)
47 \( 1 + (-0.728 + 0.684i)T \)
53 \( 1 + (0.934 + 0.356i)T \)
59 \( 1 + (-0.555 - 0.831i)T \)
61 \( 1 + (-0.998 - 0.0455i)T \)
67 \( 1 + (0.544 - 0.838i)T \)
71 \( 1 + (-0.844 - 0.536i)T \)
73 \( 1 + (0.934 - 0.356i)T \)
79 \( 1 + (-0.304 + 0.952i)T \)
83 \( 1 + (-0.851 + 0.525i)T \)
89 \( 1 + (0.454 + 0.890i)T \)
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

−21.2692531146769092306076397606, −19.87342137198866380280401266584, −19.664898186963882617369884767, −18.5905141918515660302926053486, −17.83403161172561144759487166995, −16.78339725251328589360521645700, −16.033081673701102546818144248217, −15.61081688326564048425807739917, −14.16163918970646678009663685445, −13.27634917186885599014336247652, −12.985356673542901477410281964, −12.14918275023866175752611888851, −11.605772144191436006728147136158, −10.40959410779339732559558867473, −9.89111526837925454911384726657, −8.735818166577260455130782733121, −7.48287768820559197344236273040, −6.459168823560258211076602764765, −5.85354055262962470334544924942, −5.0393247745315191069154513451, −4.42236555377411040850295952716, −2.79296079575817567353806980490, −2.253328787083316411474021325336, −0.779002472346480490877491774003, −0.16579052739836775261874194940, 2.04128927168850889363968965717, 3.00650919857112406703913543297, 4.00414580065095247877684508255, 4.693511014935786890975378281940, 5.87568824995289840261652060299, 6.36156759947526520325100723171, 7.05622064897983712494104899809, 8.03394444526001465867770550596, 9.63666990690609350375755881633, 10.25888971354635765985501670801, 10.864305903562955700244771089790, 12.09666771804356315529723612600, 12.51215699103615698879739201291, 13.52358710249482248289767568736, 14.36004889040947976301977585794, 15.29305764375771360950781676498, 15.6246982593514897591929915946, 16.699140902823267916738777959854, 17.26014139671916485560315490616, 17.95925949198373273970683600481, 19.085526748801806106620495053170, 20.03151620365275505966636941634, 21.121208716400099180874754091414, 21.63015872132787211052522099202, 22.37704678543201849797278679380

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