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Properties

Label 1-967-967.108-r1-0-0
Degree $1$
Conductor $967$
Sign $-0.610 + 0.792i$
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  + (0.692 − 0.721i)2-s + (0.957 − 0.288i)3-s + (−0.0422 − 0.999i)4-s + (−0.997 − 0.0649i)5-s + (0.454 − 0.890i)6-s + (0.347 + 0.937i)7-s + (−0.750 − 0.660i)8-s + (0.833 − 0.552i)9-s + (−0.737 + 0.675i)10-s + (0.353 + 0.935i)11-s + (−0.328 − 0.944i)12-s + (−0.633 − 0.773i)13-s + (0.917 + 0.398i)14-s + (−0.974 + 0.225i)15-s + (−0.996 + 0.0844i)16-s + (0.560 + 0.828i)17-s + ⋯
L(s)  = 1  + (0.692 − 0.721i)2-s + (0.957 − 0.288i)3-s + (−0.0422 − 0.999i)4-s + (−0.997 − 0.0649i)5-s + (0.454 − 0.890i)6-s + (0.347 + 0.937i)7-s + (−0.750 − 0.660i)8-s + (0.833 − 0.552i)9-s + (−0.737 + 0.675i)10-s + (0.353 + 0.935i)11-s + (−0.328 − 0.944i)12-s + (−0.633 − 0.773i)13-s + (0.917 + 0.398i)14-s + (−0.974 + 0.225i)15-s + (−0.996 + 0.0844i)16-s + (0.560 + 0.828i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.610 + 0.792i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.610 + 0.792i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $-0.610 + 0.792i$
Analytic conductor: \(103.918\)
Root analytic conductor: \(103.918\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (108, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (1:\ ),\ -0.610 + 0.792i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.3930647067 - 0.7990040162i\)
\(L(\frac12)\) \(\approx\) \(-0.3930647067 - 0.7990040162i\)
\(L(1)\) \(\approx\) \(1.223060214 - 0.7748229807i\)
\(L(1)\) \(\approx\) \(1.223060214 - 0.7748229807i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (0.692 - 0.721i)T \)
3 \( 1 + (0.957 - 0.288i)T \)
5 \( 1 + (-0.997 - 0.0649i)T \)
7 \( 1 + (0.347 + 0.937i)T \)
11 \( 1 + (0.353 + 0.935i)T \)
13 \( 1 + (-0.633 - 0.773i)T \)
17 \( 1 + (0.560 + 0.828i)T \)
19 \( 1 + (0.0812 - 0.996i)T \)
23 \( 1 + (-0.993 + 0.116i)T \)
29 \( 1 + (-0.0876 - 0.996i)T \)
31 \( 1 + (-0.522 - 0.852i)T \)
37 \( 1 + (-0.999 + 0.0130i)T \)
41 \( 1 + (-0.854 - 0.519i)T \)
43 \( 1 + (-0.867 - 0.497i)T \)
47 \( 1 + (0.783 + 0.620i)T \)
53 \( 1 + (-0.648 + 0.761i)T \)
59 \( 1 + (-0.999 + 0.0325i)T \)
61 \( 1 + (-0.877 + 0.480i)T \)
67 \( 1 + (0.477 + 0.878i)T \)
71 \( 1 + (0.279 - 0.960i)T \)
73 \( 1 + (-0.648 - 0.761i)T \)
79 \( 1 + (-0.914 - 0.404i)T \)
83 \( 1 + (0.969 + 0.244i)T \)
89 \( 1 + (-0.999 - 0.0260i)T \)
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

−21.988687369047480439005430355790, −21.41411486842023872643580116059, −20.385051837520777624809230809735, −20.024767226348940606039980315385, −18.981932251719336045681870368019, −18.24509796836639809453662819536, −16.72343167308079513891280568447, −16.47492016459476689680892228783, −15.73208374538409002779354883885, −14.680555247020227441576443832951, −14.16821055891181602626438911898, −13.81311948543720702927124340193, −12.55103818798715653935991736632, −11.8361319705399347054967249282, −10.90785425505863066370636526118, −9.79865375339241411639724917101, −8.65538099083550649981696874233, −8.09512320915622156502927183117, −7.31473224381736271335656346375, −6.73932164669500021469216038025, −5.19984639426254930342870864773, −4.423968111078169216707420758871, −3.59583624876547958682955214287, −3.18417389321408370860443802466, −1.60508646026735794752481305080, 0.11604715883571788395349911232, 1.5449988676405287663393786464, 2.3551268852100547160360269336, 3.22797065683489885274101634096, 4.10673180406851447701199172668, 4.87368501828824508151261234515, 5.99954376491692032211049493396, 7.19554966958197789919693101796, 7.95870574986231499812951259094, 8.89404120456119006717575807511, 9.69194441565034715234077907484, 10.59402155500792309589834538616, 11.84076253024597431242127271490, 12.21036370036013021893158367407, 12.84920544288837361749656468135, 13.86726679641524282786089098169, 14.82220665726802011652545268558, 15.23766354114616016881847529816, 15.62928279625734439008298276547, 17.32315628834066467388353778140, 18.29461792318668133497410598698, 19.01147525761420895360520240118, 19.62432298199306839793138766641, 20.25292559184062788473974765502, 20.81607470163708278003097222960

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