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Properties

Label 1-967-967.8-r0-0-0
Degree $1$
Conductor $967$
Sign $0.749 - 0.661i$
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  + (−0.799 + 0.600i)2-s + (−0.145 + 0.989i)3-s + (0.279 − 0.960i)4-s + (0.527 + 0.849i)5-s + (−0.477 − 0.878i)6-s + (0.425 − 0.905i)7-s + (0.353 + 0.935i)8-s + (−0.957 − 0.288i)9-s + (−0.932 − 0.362i)10-s + (−0.822 + 0.568i)11-s + (0.909 + 0.416i)12-s + (−0.822 − 0.568i)13-s + (0.203 + 0.979i)14-s + (−0.917 + 0.398i)15-s + (−0.844 − 0.536i)16-s + (−0.883 + 0.468i)17-s + ⋯
L(s)  = 1  + (−0.799 + 0.600i)2-s + (−0.145 + 0.989i)3-s + (0.279 − 0.960i)4-s + (0.527 + 0.849i)5-s + (−0.477 − 0.878i)6-s + (0.425 − 0.905i)7-s + (0.353 + 0.935i)8-s + (−0.957 − 0.288i)9-s + (−0.932 − 0.362i)10-s + (−0.822 + 0.568i)11-s + (0.909 + 0.416i)12-s + (−0.822 − 0.568i)13-s + (0.203 + 0.979i)14-s + (−0.917 + 0.398i)15-s + (−0.844 − 0.536i)16-s + (−0.883 + 0.468i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.749 - 0.661i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.749 - 0.661i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $0.749 - 0.661i$
Analytic conductor: \(4.49072\)
Root analytic conductor: \(4.49072\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (8, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (0:\ ),\ 0.749 - 0.661i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3018018704 - 0.1141364474i\)
\(L(\frac12)\) \(\approx\) \(0.3018018704 - 0.1141364474i\)
\(L(1)\) \(\approx\) \(0.5153221361 + 0.2654237597i\)
\(L(1)\) \(\approx\) \(0.5153221361 + 0.2654237597i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (-0.799 + 0.600i)T \)
3 \( 1 + (-0.145 + 0.989i)T \)
5 \( 1 + (0.527 + 0.849i)T \)
7 \( 1 + (0.425 - 0.905i)T \)
11 \( 1 + (-0.822 + 0.568i)T \)
13 \( 1 + (-0.822 - 0.568i)T \)
17 \( 1 + (-0.883 + 0.468i)T \)
19 \( 1 + (-0.297 - 0.954i)T \)
23 \( 1 + (-0.998 - 0.0585i)T \)
29 \( 1 + (0.737 - 0.675i)T \)
31 \( 1 + (-0.511 + 0.859i)T \)
37 \( 1 + (0.494 + 0.869i)T \)
41 \( 1 + (0.962 - 0.269i)T \)
43 \( 1 + (-0.260 + 0.965i)T \)
47 \( 1 + (0.653 - 0.756i)T \)
53 \( 1 + (-0.576 - 0.816i)T \)
59 \( 1 + (0.874 - 0.485i)T \)
61 \( 1 + (0.962 - 0.269i)T \)
67 \( 1 + (-0.511 - 0.859i)T \)
71 \( 1 + (-0.799 - 0.600i)T \)
73 \( 1 + (-0.576 + 0.816i)T \)
79 \( 1 + (-0.668 - 0.744i)T \)
83 \( 1 + (0.389 - 0.921i)T \)
89 \( 1 + (-0.511 - 0.859i)T \)
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

−21.74694737879093852540766037295, −20.874074191315868532822854194875, −20.215437615039845131302931994216, −19.327436429884038607463251954232, −18.65469878997615296083531407304, −17.94864796916259845139491927756, −17.48779005012083651496744598389, −16.4773340929773894237528383190, −15.96366408057239071306962183984, −14.48622488456927671636957329313, −13.58862387010374854792118928166, −12.76542781558868224301554557653, −12.21774161273118315966188369992, −11.55347180593349812146985261695, −10.62244125287139645851594271002, −9.49990962554287960415290706872, −8.77661930882762155771700998052, −8.14853441337872794015156238064, −7.37172894919511235465965852799, −6.156186783884842256433347202925, −5.420255089459155666011883645500, −4.25175595672116121253042124257, −2.47059732987775290282273706571, −2.276903029043568096209621596822, −1.170801522065903830283975677496, 0.186170100843853137365419535571, 2.00421907004569337960766585147, 2.87128559474394536769226825452, 4.379307502379514319713825212277, 5.03970981738774248448030201456, 6.06356214690964729211931790141, 6.92019669552161709620877913923, 7.74550940457039392210864671717, 8.635975177609984297711974371033, 9.84547859220601196940382522275, 10.14564703072411195430980042193, 10.81092581825383961672372100374, 11.5007987642424294332408633088, 13.124899486711378991778937922102, 14.10377859564503382552737019941, 14.75422129627157615980226640980, 15.37284703802680238177262907985, 16.08639402389675482238981368613, 17.16341043285714752887039415326, 17.677469586767069499406450378639, 17.996949145622791496852957091664, 19.434522637193816426774104846029, 19.99778650630788509497171737698, 20.75910886901738474361061283824, 21.71517149887971054130154681822

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