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Properties

Label 1-967-967.916-r0-0-0
Degree $1$
Conductor $967$
Sign $0.605 + 0.796i$
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  + (−0.917 − 0.398i)2-s + (0.460 − 0.887i)3-s + (0.682 + 0.730i)4-s + (−0.0682 + 0.997i)5-s + (−0.775 + 0.631i)6-s + (−0.990 + 0.136i)7-s + (−0.334 − 0.942i)8-s + (−0.576 − 0.816i)9-s + (0.460 − 0.887i)10-s + (−0.576 − 0.816i)11-s + (0.962 − 0.269i)12-s + (−0.576 + 0.816i)13-s + (0.962 + 0.269i)14-s + (0.854 + 0.519i)15-s + (−0.0682 + 0.997i)16-s + (0.854 − 0.519i)17-s + ⋯
L(s)  = 1  + (−0.917 − 0.398i)2-s + (0.460 − 0.887i)3-s + (0.682 + 0.730i)4-s + (−0.0682 + 0.997i)5-s + (−0.775 + 0.631i)6-s + (−0.990 + 0.136i)7-s + (−0.334 − 0.942i)8-s + (−0.576 − 0.816i)9-s + (0.460 − 0.887i)10-s + (−0.576 − 0.816i)11-s + (0.962 − 0.269i)12-s + (−0.576 + 0.816i)13-s + (0.962 + 0.269i)14-s + (0.854 + 0.519i)15-s + (−0.0682 + 0.997i)16-s + (0.854 − 0.519i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.605 + 0.796i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.605 + 0.796i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $0.605 + 0.796i$
Analytic conductor: \(4.49072\)
Root analytic conductor: \(4.49072\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (916, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (0:\ ),\ 0.605 + 0.796i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4816767912 + 0.2388906979i\)
\(L(\frac12)\) \(\approx\) \(0.4816767912 + 0.2388906979i\)
\(L(1)\) \(\approx\) \(0.6151125129 - 0.1108819305i\)
\(L(1)\) \(\approx\) \(0.6151125129 - 0.1108819305i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (-0.917 - 0.398i)T \)
3 \( 1 + (0.460 - 0.887i)T \)
5 \( 1 + (-0.0682 + 0.997i)T \)
7 \( 1 + (-0.990 + 0.136i)T \)
11 \( 1 + (-0.576 - 0.816i)T \)
13 \( 1 + (-0.576 + 0.816i)T \)
17 \( 1 + (0.854 - 0.519i)T \)
19 \( 1 + (0.460 - 0.887i)T \)
23 \( 1 + (0.682 + 0.730i)T \)
29 \( 1 + (-0.576 + 0.816i)T \)
31 \( 1 + (-0.334 - 0.942i)T \)
37 \( 1 + (-0.576 + 0.816i)T \)
41 \( 1 + (-0.775 + 0.631i)T \)
43 \( 1 + (0.854 + 0.519i)T \)
47 \( 1 + (0.854 + 0.519i)T \)
53 \( 1 + (0.682 + 0.730i)T \)
59 \( 1 + (0.682 - 0.730i)T \)
61 \( 1 + (-0.775 + 0.631i)T \)
67 \( 1 + (-0.334 + 0.942i)T \)
71 \( 1 + (-0.917 + 0.398i)T \)
73 \( 1 + (0.682 - 0.730i)T \)
79 \( 1 + (0.682 - 0.730i)T \)
83 \( 1 + (-0.775 + 0.631i)T \)
89 \( 1 + (-0.334 + 0.942i)T \)
97 \( 1 + T \)
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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

−21.22294417399012756620179971700, −20.656291782336037436422051505, −20.03985266699321064313199969685, −19.42682597299246262579228367325, −18.59736800001524816819175091473, −17.365017082832788678779432509168, −16.82329715487970267751172463792, −16.17618722613214172123400458185, −15.49480486774832801425166526732, −14.885678397172339940812971791474, −13.85772998595480331942047475245, −12.70262398851908781183002286533, −12.105678875910648054158613742202, −10.54946997889676792253538040403, −10.16723378847102846123320579061, −9.4681464860880179999279083292, −8.71888665771568329320610076917, −7.883596915315664795599706758, −7.18625412094311068622973336579, −5.64201453743892340816060563170, −5.29304858327433663771077265434, −4.01882638177713281186515329504, −2.952064586151382672277533865317, −1.86023586262171625647741472552, −0.32636730967715044485398536947, 1.067250095622625244915820750578, 2.43873466848127929395741434377, 2.94710158459719180346787451622, 3.6136027176986780010446038449, 5.66246288328795570704700854905, 6.64326580922978183522961549855, 7.23202608064570161951026767898, 7.81728473008699326047406268114, 9.07932948839115227494542393831, 9.49275387482447146896735422441, 10.5422661564452830805681292829, 11.48345818427260900026298130481, 12.01819005126477402625295092587, 13.121769568532820660107365099905, 13.675570778856164687850538060115, 14.76219373491017793481813742720, 15.62545762013288817107087977385, 16.50548258287853733198230560570, 17.34200360885596390776552556280, 18.359273809779638712314268715952, 18.79357114048952895839193823957, 19.22973764434052937532767203466, 19.934162832644524623128120955028, 20.90906056198237184459679381633, 21.8115092989009455496283208898

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