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Properties

Label 1-967-967.126-r1-0-0
Degree $1$
Conductor $967$
Sign $-0.324 + 0.945i$
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  + (−0.0292 + 0.999i)2-s + (−0.996 + 0.0779i)3-s + (−0.998 − 0.0585i)4-s + (0.945 − 0.325i)5-s + (−0.0487 − 0.998i)6-s + (−0.972 + 0.232i)7-s + (0.0876 − 0.996i)8-s + (0.987 − 0.155i)9-s + (0.297 + 0.954i)10-s + (0.737 + 0.675i)11-s + (0.999 − 0.0195i)12-s + (−0.737 + 0.675i)13-s + (−0.203 − 0.979i)14-s + (−0.917 + 0.398i)15-s + (0.993 + 0.116i)16-s + (−0.260 − 0.965i)17-s + ⋯
L(s)  = 1  + (−0.0292 + 0.999i)2-s + (−0.996 + 0.0779i)3-s + (−0.998 − 0.0585i)4-s + (0.945 − 0.325i)5-s + (−0.0487 − 0.998i)6-s + (−0.972 + 0.232i)7-s + (0.0876 − 0.996i)8-s + (0.987 − 0.155i)9-s + (0.297 + 0.954i)10-s + (0.737 + 0.675i)11-s + (0.999 − 0.0195i)12-s + (−0.737 + 0.675i)13-s + (−0.203 − 0.979i)14-s + (−0.917 + 0.398i)15-s + (0.993 + 0.116i)16-s + (−0.260 − 0.965i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.324 + 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.324 + 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $-0.324 + 0.945i$
Analytic conductor: \(103.918\)
Root analytic conductor: \(103.918\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (126, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (1:\ ),\ -0.324 + 0.945i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7079403802 + 0.9909502414i\)
\(L(\frac12)\) \(\approx\) \(0.7079403802 + 0.9909502414i\)
\(L(1)\) \(\approx\) \(0.6691889141 + 0.3959596786i\)
\(L(1)\) \(\approx\) \(0.6691889141 + 0.3959596786i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (-0.0292 + 0.999i)T \)
3 \( 1 + (-0.996 + 0.0779i)T \)
5 \( 1 + (0.945 - 0.325i)T \)
7 \( 1 + (-0.972 + 0.232i)T \)
11 \( 1 + (0.737 + 0.675i)T \)
13 \( 1 + (-0.737 + 0.675i)T \)
17 \( 1 + (-0.260 - 0.965i)T \)
19 \( 1 + (0.932 + 0.362i)T \)
23 \( 1 + (-0.279 + 0.960i)T \)
29 \( 1 + (0.822 + 0.568i)T \)
31 \( 1 + (-0.724 - 0.689i)T \)
37 \( 1 + (0.371 - 0.928i)T \)
41 \( 1 + (-0.962 + 0.269i)T \)
43 \( 1 + (0.883 + 0.468i)T \)
47 \( 1 + (-0.999 - 0.0390i)T \)
53 \( 1 + (-0.576 - 0.816i)T \)
59 \( 1 + (0.165 - 0.986i)T \)
61 \( 1 + (0.962 - 0.269i)T \)
67 \( 1 + (0.724 - 0.689i)T \)
71 \( 1 + (-0.0292 - 0.999i)T \)
73 \( 1 + (-0.576 + 0.816i)T \)
79 \( 1 + (-0.924 - 0.380i)T \)
83 \( 1 + (0.811 + 0.584i)T \)
89 \( 1 + (0.724 - 0.689i)T \)
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

−21.658570685204618544044920286903, −20.54393974935111198736716879767, −19.668630201904415704092125527784, −18.978215203478709621977781024242, −18.20272537168294650320064343479, −17.39177080659747544783158336659, −16.986805383243610194374607503648, −16.03491769868144134265281158989, −14.75746756573175882735540254508, −13.82011711248122694062713126418, −13.11593924158222825469591495034, −12.47275362613772013112689765177, −11.67670178772919395323812797431, −10.67946684414010142234847552147, −10.17988496664520076175781501117, −9.57005202260771417933484189091, −8.54564831344208261674041322040, −7.11791978394251311097722231482, −6.23586232616979376968744880380, −5.57705167755959080342674835443, −4.55957803933085111882583224095, −3.467002289073940316672473004938, −2.5775490675350518290371127945, −1.36250305774175295144553327647, −0.48912144779864456387874223946, 0.685118029661817032915252369777, 1.8882181392248225745661091396, 3.555350375942288763622479145421, 4.71067245043311403312450081319, 5.29192716963772737814580571584, 6.18312763508257863471979588957, 6.78600532460906989855649264599, 7.478912885578718348601440086710, 9.09774393486914445707482979515, 9.667135687583199473209258472576, 9.92931955767966951906988876181, 11.51063234649075331290402369803, 12.38539276921429280499565774255, 13.001728846444085314365041797695, 13.915026245149984036230628740814, 14.666914456533665398533251378797, 15.87547969247783787962154064535, 16.26976190064795068556238343871, 16.994060759767691550284377766902, 17.704469372239120739246626955, 18.23139671974504601942770052692, 19.17688959892632599075754125259, 20.19055495610936504278677900119, 21.47901179837611834849885620400, 22.07941657236850223463635162253

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