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Properties

Label 1-967-967.65-r0-0-0
Degree $1$
Conductor $967$
Sign $0.559 - 0.828i$
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  + (−0.0162 + 0.999i)2-s + (0.737 − 0.675i)3-s + (−0.999 − 0.0325i)4-s + (0.861 − 0.508i)5-s + (0.663 + 0.748i)6-s + (−0.383 + 0.923i)7-s + (0.0487 − 0.998i)8-s + (0.0876 − 0.996i)9-s + (0.494 + 0.869i)10-s + (−0.724 − 0.689i)11-s + (−0.759 + 0.651i)12-s + (0.959 + 0.282i)13-s + (−0.917 − 0.398i)14-s + (0.291 − 0.956i)15-s + (0.997 + 0.0649i)16-s + (−0.145 − 0.989i)17-s + ⋯
L(s)  = 1  + (−0.0162 + 0.999i)2-s + (0.737 − 0.675i)3-s + (−0.999 − 0.0325i)4-s + (0.861 − 0.508i)5-s + (0.663 + 0.748i)6-s + (−0.383 + 0.923i)7-s + (0.0487 − 0.998i)8-s + (0.0876 − 0.996i)9-s + (0.494 + 0.869i)10-s + (−0.724 − 0.689i)11-s + (−0.759 + 0.651i)12-s + (0.959 + 0.282i)13-s + (−0.917 − 0.398i)14-s + (0.291 − 0.956i)15-s + (0.997 + 0.0649i)16-s + (−0.145 − 0.989i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.559 - 0.828i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.559 - 0.828i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $0.559 - 0.828i$
Analytic conductor: \(4.49072\)
Root analytic conductor: \(4.49072\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (0:\ ),\ 0.559 - 0.828i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.325480038 - 0.7046699848i\)
\(L(\frac12)\) \(\approx\) \(1.325480038 - 0.7046699848i\)
\(L(1)\) \(\approx\) \(1.202817007 + 0.008014962410i\)
\(L(1)\) \(\approx\) \(1.202817007 + 0.008014962410i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (-0.0162 + 0.999i)T \)
3 \( 1 + (0.737 - 0.675i)T \)
5 \( 1 + (0.861 - 0.508i)T \)
7 \( 1 + (-0.383 + 0.923i)T \)
11 \( 1 + (-0.724 - 0.689i)T \)
13 \( 1 + (0.959 + 0.282i)T \)
17 \( 1 + (-0.145 - 0.989i)T \)
19 \( 1 + (-0.618 - 0.785i)T \)
23 \( 1 + (-0.945 - 0.325i)T \)
29 \( 1 + (-0.511 - 0.859i)T \)
31 \( 1 + (-0.911 - 0.410i)T \)
37 \( 1 + (-0.741 - 0.670i)T \)
41 \( 1 + (0.854 + 0.519i)T \)
43 \( 1 + (-0.430 + 0.902i)T \)
47 \( 1 + (0.152 + 0.988i)T \)
53 \( 1 + (0.983 + 0.181i)T \)
59 \( 1 + (-0.964 - 0.263i)T \)
61 \( 1 + (0.0227 - 0.999i)T \)
67 \( 1 + (0.811 + 0.584i)T \)
71 \( 1 + (0.874 + 0.485i)T \)
73 \( 1 + (0.983 - 0.181i)T \)
79 \( 1 + (-0.0422 + 0.999i)T \)
83 \( 1 + (-0.171 + 0.985i)T \)
89 \( 1 + (0.100 - 0.994i)T \)
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

−21.59500572180995012677774831165, −21.05530633257177280817477986385, −20.32322286260044132169664344749, −19.83712627330479966501147566377, −18.80212201019751561513256831106, −18.178079660694339635498473010762, −17.26038034247070641848279811093, −16.48071166374084923558269484065, −15.30408836409978435671917253, −14.52523078020192558110973984258, −13.74242496304727937661384852326, −13.23742053910711232315035120097, −12.47805210496681676590163882751, −10.87619940950910635880110270503, −10.459802283069229303465378328586, −10.08800660106304110926208728240, −9.11094098796780922409715781614, −8.28507002115677382793717476458, −7.31026805226715258201805073747, −5.95310335615416628230396028519, −5.02212209117402872960012801713, −3.73978144003104592943908860520, −3.56466638261708660779957756986, −2.21987813778737067122924146863, −1.62700342096041417477427644703, 0.56412509582653546831766718406, 1.992468358804639987848922209550, 2.896299616461416844460406205643, 4.12330013251348339548372957702, 5.36806552106113567557641779404, 6.04634743699147637447321090984, 6.65191458118037389538739924877, 7.855429561962534150315818357514, 8.555740339309631083378325905046, 9.17739144301911777912633109486, 9.7379843040893871891294601864, 11.22164955390211813317816541627, 12.58623602984541165042831006422, 13.0238886623188765322785420806, 13.76277100722198517434248485677, 14.28889836243341093215326497037, 15.45828591203414354033735392438, 15.937641194841566861585412630545, 16.8151864891735054983709934412, 17.95786151179633014744033146624, 18.30673124170249510382251099201, 18.93224090603575538430582000682, 19.95421337986945737901329120686, 21.07305988727931399901194186748, 21.50406752298712173484587043464

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