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Properties

Label 1-967-967.121-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: $\chi_{967} (121, \cdot )$
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

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

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