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Properties

Label 1-967-967.104-r1-0-0
Degree $1$
Conductor $967$
Sign $0.989 + 0.142i$
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  + (−0.0812 + 0.996i)2-s + (0.844 − 0.536i)3-s + (−0.986 − 0.161i)4-s + (0.889 + 0.457i)5-s + (0.465 + 0.884i)6-s + (0.922 + 0.386i)7-s + (0.241 − 0.970i)8-s + (0.425 − 0.905i)9-s + (−0.527 + 0.849i)10-s + (0.787 + 0.615i)11-s + (−0.919 + 0.392i)12-s + (−0.139 − 0.990i)13-s + (−0.460 + 0.887i)14-s + (0.995 − 0.0909i)15-s + (0.947 + 0.319i)16-s + (−0.668 − 0.744i)17-s + ⋯
L(s)  = 1  + (−0.0812 + 0.996i)2-s + (0.844 − 0.536i)3-s + (−0.986 − 0.161i)4-s + (0.889 + 0.457i)5-s + (0.465 + 0.884i)6-s + (0.922 + 0.386i)7-s + (0.241 − 0.970i)8-s + (0.425 − 0.905i)9-s + (−0.527 + 0.849i)10-s + (0.787 + 0.615i)11-s + (−0.919 + 0.392i)12-s + (−0.139 − 0.990i)13-s + (−0.460 + 0.887i)14-s + (0.995 − 0.0909i)15-s + (0.947 + 0.319i)16-s + (−0.668 − 0.744i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.989 + 0.142i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.989 + 0.142i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $0.989 + 0.142i$
Analytic conductor: \(103.918\)
Root analytic conductor: \(103.918\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (104, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (1:\ ),\ 0.989 + 0.142i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.738179300 + 0.2684913190i\)
\(L(\frac12)\) \(\approx\) \(3.738179300 + 0.2684913190i\)
\(L(1)\) \(\approx\) \(1.666138120 + 0.3946348009i\)
\(L(1)\) \(\approx\) \(1.666138120 + 0.3946348009i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (-0.0812 + 0.996i)T \)
3 \( 1 + (0.844 - 0.536i)T \)
5 \( 1 + (0.889 + 0.457i)T \)
7 \( 1 + (0.922 + 0.386i)T \)
11 \( 1 + (0.787 + 0.615i)T \)
13 \( 1 + (-0.139 - 0.990i)T \)
17 \( 1 + (-0.668 - 0.744i)T \)
19 \( 1 + (-0.190 - 0.981i)T \)
23 \( 1 + (-0.0876 + 0.996i)T \)
29 \( 1 + (0.442 - 0.896i)T \)
31 \( 1 + (0.516 - 0.856i)T \)
37 \( 1 + (-0.861 - 0.508i)T \)
41 \( 1 + (0.917 - 0.398i)T \)
43 \( 1 + (0.791 + 0.610i)T \)
47 \( 1 + (-0.692 - 0.721i)T \)
53 \( 1 + (0.613 + 0.789i)T \)
59 \( 1 + (-0.235 - 0.971i)T \)
61 \( 1 + (0.113 - 0.993i)T \)
67 \( 1 + (0.999 - 0.0195i)T \)
71 \( 1 + (-0.822 + 0.568i)T \)
73 \( 1 + (0.613 - 0.789i)T \)
79 \( 1 + (0.209 - 0.977i)T \)
83 \( 1 + (-0.759 + 0.651i)T \)
89 \( 1 + (-0.483 + 0.875i)T \)
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

−21.23398167578213695156482389013, −20.96641390214550268167490107884, −20.044570289193644919477036052984, −19.442412103614560211105827223912, −18.5887789651553318616170057159, −17.65187479746870159455984418719, −16.90233604742867001666045579604, −16.26243886731346347566807775863, −14.619909752239791377050793873, −14.267048526131387749851489627072, −13.729637955847801395221762180207, −12.7705336523925610517178183784, −11.878224418455967223434506529566, −10.77387068725345647640135640040, −10.337182608175531691282604797101, −9.33734719956259165289405026501, −8.63870220930825975370200213197, −8.26578744343934484781083874917, −6.722617074955491512328022127415, −5.41161908842087097368494636824, −4.42184367512077436335821718452, −4.01874990385695412563242747039, −2.73322216553499711247906324459, −1.76918930867962112992385028632, −1.24447350266621758136104699386, 0.75198044384089740268633830296, 1.8956594855351820347010853654, 2.774165972278195687370489690976, 4.108126364813966104378695263956, 5.09615921562089242887832455929, 6.03357008695831698516585724474, 6.90032830690305865934013698215, 7.553408528002965158377170904360, 8.413619642353823224492814905572, 9.31935157614484225912000220845, 9.71572090325088251289301123185, 11.03586004752926650627531051517, 12.21925657193485558823357752628, 13.19610875119128256256839038002, 13.80663135568922123292329215595, 14.434073052819937796362832862460, 15.21208089745775800261692619983, 15.59645884099915078358541556364, 17.37098157625839129102064726248, 17.5570277577338537688824320239, 18.1142556760426210033573399192, 19.05506413515177775628617782621, 19.86232738878802993063699977132, 20.83841738837433372497983068394, 21.64976298877479576001689711579

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