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Properties

Label 1-967-967.141-r0-0-0
Degree $1$
Conductor $967$
Sign $-0.865 + 0.500i$
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  + (0.959 + 0.282i)2-s + (−0.442 + 0.896i)3-s + (0.840 + 0.541i)4-s + (−0.272 − 0.962i)5-s + (−0.677 + 0.735i)6-s + (−0.322 + 0.946i)7-s + (0.653 + 0.756i)8-s + (−0.608 − 0.793i)9-s + (0.00975 − 0.999i)10-s + (0.909 − 0.416i)11-s + (−0.857 + 0.514i)12-s + (−0.815 + 0.579i)13-s + (−0.576 + 0.816i)14-s + (0.983 + 0.181i)15-s + (0.413 + 0.910i)16-s + (−0.844 − 0.536i)17-s + ⋯
L(s)  = 1  + (0.959 + 0.282i)2-s + (−0.442 + 0.896i)3-s + (0.840 + 0.541i)4-s + (−0.272 − 0.962i)5-s + (−0.677 + 0.735i)6-s + (−0.322 + 0.946i)7-s + (0.653 + 0.756i)8-s + (−0.608 − 0.793i)9-s + (0.00975 − 0.999i)10-s + (0.909 − 0.416i)11-s + (−0.857 + 0.514i)12-s + (−0.815 + 0.579i)13-s + (−0.576 + 0.816i)14-s + (0.983 + 0.181i)15-s + (0.413 + 0.910i)16-s + (−0.844 − 0.536i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.865 + 0.500i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.865 + 0.500i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $-0.865 + 0.500i$
Analytic conductor: \(4.49072\)
Root analytic conductor: \(4.49072\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (141, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (0:\ ),\ -0.865 + 0.500i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4275371311 + 1.594959580i\)
\(L(\frac12)\) \(\approx\) \(0.4275371311 + 1.594959580i\)
\(L(1)\) \(\approx\) \(1.145476524 + 0.7591518814i\)
\(L(1)\) \(\approx\) \(1.145476524 + 0.7591518814i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (0.959 + 0.282i)T \)
3 \( 1 + (-0.442 + 0.896i)T \)
5 \( 1 + (-0.272 - 0.962i)T \)
7 \( 1 + (-0.322 + 0.946i)T \)
11 \( 1 + (0.909 - 0.416i)T \)
13 \( 1 + (-0.815 + 0.579i)T \)
17 \( 1 + (-0.844 - 0.536i)T \)
19 \( 1 + (0.672 + 0.739i)T \)
23 \( 1 + (-0.477 + 0.878i)T \)
29 \( 1 + (-0.999 + 0.0195i)T \)
31 \( 1 + (0.401 + 0.915i)T \)
37 \( 1 + (-0.0552 + 0.998i)T \)
41 \( 1 + (0.682 + 0.730i)T \)
43 \( 1 + (-0.597 - 0.801i)T \)
47 \( 1 + (0.471 + 0.881i)T \)
53 \( 1 + (-0.247 - 0.968i)T \)
59 \( 1 + (0.602 + 0.797i)T \)
61 \( 1 + (-0.974 + 0.225i)T \)
67 \( 1 + (0.592 + 0.805i)T \)
71 \( 1 + (-0.724 - 0.689i)T \)
73 \( 1 + (-0.247 + 0.968i)T \)
79 \( 1 + (-0.197 - 0.980i)T \)
83 \( 1 + (0.867 + 0.497i)T \)
89 \( 1 + (-0.993 + 0.110i)T \)
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

−21.95073272044931321449053315202, −20.3816282503353586897277357284, −19.790751961143190867365947112773, −19.39591121377342126587602890806, −18.37949820997716270316441504449, −17.454989314162073077845159407008, −16.79449577276281608902711177517, −15.69036426109378415083182957817, −14.79137811467673782101871683080, −14.16190957804879604495345634955, −13.40548310065347949189094430980, −12.65616809995432039615620194806, −11.88009595563654502029074264974, −11.0841148754966943167767887248, −10.544230338318803087516840412499, −9.54915721851959107173547271554, −7.78747367995355814940881263410, −7.16349291265459786051222531410, −6.609868499327068762350274233993, −5.844803752970025394793012471198, −4.59144786984751479518380892051, −3.79412022187927130601192622742, −2.70525613146996026071774738675, −1.91363940451873906592137944753, −0.512357132192916254872959000524, 1.60870010061561043126927203485, 2.96875782760433313503583543354, 3.86813850198194197210391823664, 4.640807248410142400862962314221, 5.39598612310571657811571315162, 6.0432540648729504623964168793, 7.0485490529529338722335834403, 8.32277396381260535237006946754, 9.14693610292223096924962976219, 9.79268843722874110395930328089, 11.27523611529488474611478117597, 11.87076450658995988589390137784, 12.19463398914996180678490027411, 13.33094785487331051762394391409, 14.2617436571116565325125692575, 15.05950441490913062938989608840, 15.822557603786086611656347398293, 16.32456971418951278261814891728, 16.95436570277058851449915298174, 17.80061980093908020429629567983, 19.26412413065958339702060314184, 19.99371609305901524969875014440, 20.72759123196197594228047376650, 21.52760431825934395933689851606, 22.12777337083786642751496677758

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