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Properties

Label 1-967-967.455-r1-0-0
Degree $1$
Conductor $967$
Sign $-0.936 - 0.349i$
Analytic cond. $103.918$
Root an. cond. $103.918$
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  + (0.353 + 0.935i)2-s + (−0.425 + 0.905i)3-s + (−0.750 + 0.660i)4-s + (0.995 − 0.0974i)5-s + (−0.996 − 0.0779i)6-s + (0.967 − 0.250i)7-s + (−0.883 − 0.468i)8-s + (−0.638 − 0.769i)9-s + (0.442 + 0.896i)10-s + (0.241 + 0.970i)11-s + (−0.279 − 0.960i)12-s + (−0.241 + 0.970i)13-s + (0.576 + 0.816i)14-s + (−0.334 + 0.942i)15-s + (0.126 − 0.991i)16-s + (−0.107 + 0.994i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.935i)2-s + (−0.425 + 0.905i)3-s + (−0.750 + 0.660i)4-s + (0.995 − 0.0974i)5-s + (−0.996 − 0.0779i)6-s + (0.967 − 0.250i)7-s + (−0.883 − 0.468i)8-s + (−0.638 − 0.769i)9-s + (0.442 + 0.896i)10-s + (0.241 + 0.970i)11-s + (−0.279 − 0.960i)12-s + (−0.241 + 0.970i)13-s + (0.576 + 0.816i)14-s + (−0.334 + 0.942i)15-s + (0.126 − 0.991i)16-s + (−0.107 + 0.994i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.936 - 0.349i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.936 - 0.349i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $-0.936 - 0.349i$
Analytic conductor: \(103.918\)
Root analytic conductor: \(103.918\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (455, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (1:\ ),\ -0.936 - 0.349i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.4686539850 + 2.597520380i\)
\(L(\frac12)\) \(\approx\) \(-0.4686539850 + 2.597520380i\)
\(L(1)\) \(\approx\) \(0.7791974547 + 1.116128448i\)
\(L(1)\) \(\approx\) \(0.7791974547 + 1.116128448i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (0.353 + 0.935i)T \)
3 \( 1 + (-0.425 + 0.905i)T \)
5 \( 1 + (0.995 - 0.0974i)T \)
7 \( 1 + (0.967 - 0.250i)T \)
11 \( 1 + (0.241 + 0.970i)T \)
13 \( 1 + (-0.241 + 0.970i)T \)
17 \( 1 + (-0.107 + 0.994i)T \)
19 \( 1 + (-0.787 - 0.615i)T \)
23 \( 1 + (0.984 + 0.174i)T \)
29 \( 1 + (0.608 + 0.793i)T \)
31 \( 1 + (0.999 + 0.0390i)T \)
37 \( 1 + (0.999 + 0.0195i)T \)
41 \( 1 + (-0.682 + 0.730i)T \)
43 \( 1 + (-0.710 + 0.703i)T \)
47 \( 1 + (0.844 + 0.536i)T \)
53 \( 1 + (0.962 - 0.269i)T \)
59 \( 1 + (0.0487 - 0.998i)T \)
61 \( 1 + (0.682 - 0.730i)T \)
67 \( 1 + (-0.999 + 0.0390i)T \)
71 \( 1 + (0.353 - 0.935i)T \)
73 \( 1 + (0.962 + 0.269i)T \)
79 \( 1 + (-0.811 + 0.584i)T \)
83 \( 1 + (-0.932 + 0.362i)T \)
89 \( 1 + (-0.999 + 0.0390i)T \)
97 \( 1 + (0.623 + 0.781i)T \)
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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

−21.14226372006152929847981441744, −20.488019838952746530825182004253, −19.4789181306395141842111923940, −18.61845065865771721681656709704, −18.23867276223740264602730132677, −17.3651610426261927073269295803, −16.87793633992363238415925680926, −15.23323894615710667326934599409, −14.38104768312539744525287746127, −13.668902816195480387135677352688, −13.218889197235811563327035757976, −12.19153462034705049962063749170, −11.58563140063652098500798337626, −10.7483879131170939357308912395, −10.13088251852931625807273247368, −8.810357378580224989877398090047, −8.29032107588082297463331370083, −6.95011676451827796363861527513, −5.81678565312917661395571351398, −5.48149916907358837314397189830, −4.49205608416377570291310110968, −2.86919272270840006997499285507, −2.361686691742974465622793482861, −1.26343779983160674656890140168, −0.57464172760721685730066905573, 1.242066688913391182430282514820, 2.58016320767116084584489579067, 4.04671641043028032466260713751, 4.71532888354499763756080636657, 5.17201025566185909276095063129, 6.40130048199634519478924716131, 6.81165068627090788819906233874, 8.2195551856842792583142310606, 9.001202161118312263107780023777, 9.70071845040182080634766330519, 10.59396220487079651408867894264, 11.57075016607189495846065931619, 12.529768634694357231020498964668, 13.40958572495828575708803577213, 14.45513433075069642233525037094, 14.75141567345135504418662754936, 15.5329893319286513237593670314, 16.73524039147705261359090906408, 17.12263133895161260999261889127, 17.59450974351602396827181953298, 18.39946404957722692353673581419, 19.83222737357984882773373121111, 20.888892538484206131418351946460, 21.521927503175665550287480394425, 21.75036530114448559851423285473

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