Properties

Label 1-967-967.127-r0-0-0
Degree $1$
Conductor $967$
Sign $-0.684 - 0.728i$
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  + (−0.993 + 0.110i)2-s + (−0.799 − 0.600i)3-s + (0.975 − 0.219i)4-s + (0.989 − 0.142i)5-s + (0.861 + 0.508i)6-s + (−0.986 + 0.161i)7-s + (−0.945 + 0.325i)8-s + (0.279 + 0.960i)9-s + (−0.967 + 0.250i)10-s + (0.165 − 0.986i)11-s + (−0.911 − 0.410i)12-s + (0.771 − 0.636i)13-s + (0.962 − 0.269i)14-s + (−0.877 − 0.480i)15-s + (0.903 − 0.428i)16-s + (−0.544 − 0.838i)17-s + ⋯
L(s)  = 1  + (−0.993 + 0.110i)2-s + (−0.799 − 0.600i)3-s + (0.975 − 0.219i)4-s + (0.989 − 0.142i)5-s + (0.861 + 0.508i)6-s + (−0.986 + 0.161i)7-s + (−0.945 + 0.325i)8-s + (0.279 + 0.960i)9-s + (−0.967 + 0.250i)10-s + (0.165 − 0.986i)11-s + (−0.911 − 0.410i)12-s + (0.771 − 0.636i)13-s + (0.962 − 0.269i)14-s + (−0.877 − 0.480i)15-s + (0.903 − 0.428i)16-s + (−0.544 − 0.838i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.684 - 0.728i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.684 - 0.728i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $-0.684 - 0.728i$
Analytic conductor: \(4.49072\)
Root analytic conductor: \(4.49072\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (0:\ ),\ -0.684 - 0.728i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2084073473 - 0.4817322119i\)
\(L(\frac12)\) \(\approx\) \(0.2084073473 - 0.4817322119i\)
\(L(1)\) \(\approx\) \(0.5252741488 - 0.1850565962i\)
\(L(1)\) \(\approx\) \(0.5252741488 - 0.1850565962i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (-0.993 + 0.110i)T \)
3 \( 1 + (-0.799 - 0.600i)T \)
5 \( 1 + (0.989 - 0.142i)T \)
7 \( 1 + (-0.986 + 0.161i)T \)
11 \( 1 + (0.165 - 0.986i)T \)
13 \( 1 + (0.771 - 0.636i)T \)
17 \( 1 + (-0.544 - 0.838i)T \)
19 \( 1 + (0.177 + 0.984i)T \)
23 \( 1 + (-0.932 + 0.362i)T \)
29 \( 1 + (0.874 + 0.485i)T \)
31 \( 1 + (-0.964 - 0.263i)T \)
37 \( 1 + (-0.791 - 0.610i)T \)
41 \( 1 + (-0.775 - 0.631i)T \)
43 \( 1 + (0.413 - 0.910i)T \)
47 \( 1 + (0.663 - 0.748i)T \)
53 \( 1 + (0.291 + 0.956i)T \)
59 \( 1 + (-0.997 - 0.0714i)T \)
61 \( 1 + (-0.158 + 0.987i)T \)
67 \( 1 + (0.710 + 0.703i)T \)
71 \( 1 + (0.592 - 0.805i)T \)
73 \( 1 + (0.291 - 0.956i)T \)
79 \( 1 + (-0.566 - 0.824i)T \)
83 \( 1 + (-0.996 - 0.0844i)T \)
89 \( 1 + (0.254 - 0.967i)T \)
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

−21.85503594661224812550054734279, −21.34571321049586806281474454563, −20.36524657536870440818099571116, −19.77475626098781182614624959811, −18.62505788318713598800717807174, −17.98558727735769834610962222428, −17.304113014724493251749058861758, −16.77091549506347408956618320440, −15.851335018119702703530119688318, −15.382507661199628270750493811628, −14.18388381150503173529875397673, −12.97877413501646225103380002962, −12.34450123788927057980192473879, −11.27551433716136860188950532656, −10.57501015031071119625745407994, −9.82086720728510518708446018387, −9.42288972096915342119897471972, −8.51815823484285949610366050883, −6.87712171899632997363330195658, −6.57820192793843347037586852483, −5.826090415120566872889102351135, −4.51348288690107628000382729198, −3.44832883874708317796255614348, −2.26315347053736857006931604925, −1.23525609977094959433929924100, 0.37694622816263847379599422060, 1.42516410436397703123045216444, 2.40435433607643177050558197918, 3.49777052133633158264415823850, 5.50345169010065678519651296014, 5.83834491085223395309379473875, 6.57056221053926969048991249104, 7.41291795378318134024673312538, 8.563899342746017010840276082197, 9.20357495266515809661257011313, 10.34236242133123505288100128442, 10.623044652203810028913143769182, 11.8130838790855382683129355279, 12.46904386698007896039947948071, 13.49164407528857679516919725504, 14.04610678404722710929404019935, 15.6672960417004534393711148481, 16.180740091201918598664781168577, 16.81366453881850766634993990481, 17.59268603927863482581635003279, 18.46880936976856677927909913799, 18.608407779872017672457832922868, 19.74887927679577435941285581, 20.4560264485839829164806336135, 21.542882876169886359728656848251

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