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Properties

Label 1-6145-6145.1017-r1-0-0
Degree $1$
Conductor $6145$
Sign $0.940 - 0.338i$
Analytic cond. $660.371$
Root an. cond. $660.371$
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  + (0.951 − 0.307i)2-s + (0.957 + 0.287i)3-s + (0.811 − 0.584i)4-s + (0.999 − 0.0204i)6-s + (−0.998 + 0.0562i)7-s + (0.592 − 0.805i)8-s + (0.834 + 0.550i)9-s + (−0.0255 + 0.999i)11-s + (0.945 − 0.326i)12-s + (−0.730 − 0.683i)13-s + (−0.932 + 0.360i)14-s + (0.316 − 0.948i)16-s + (0.995 + 0.0970i)17-s + (0.963 + 0.267i)18-s + (0.494 + 0.869i)19-s + ⋯
L(s)  = 1  + (0.951 − 0.307i)2-s + (0.957 + 0.287i)3-s + (0.811 − 0.584i)4-s + (0.999 − 0.0204i)6-s + (−0.998 + 0.0562i)7-s + (0.592 − 0.805i)8-s + (0.834 + 0.550i)9-s + (−0.0255 + 0.999i)11-s + (0.945 − 0.326i)12-s + (−0.730 − 0.683i)13-s + (−0.932 + 0.360i)14-s + (0.316 − 0.948i)16-s + (0.995 + 0.0970i)17-s + (0.963 + 0.267i)18-s + (0.494 + 0.869i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6145 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.940 - 0.338i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6145 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.940 - 0.338i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6145\)    =    \(5 \cdot 1229\)
Sign: $0.940 - 0.338i$
Analytic conductor: \(660.371\)
Root analytic conductor: \(660.371\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6145} (1017, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6145,\ (1:\ ),\ 0.940 - 0.338i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(6.960178230 - 1.213583987i\)
\(L(\frac12)\) \(\approx\) \(6.960178230 - 1.213583987i\)
\(L(1)\) \(\approx\) \(2.535509877 - 0.2675631950i\)
\(L(1)\) \(\approx\) \(2.535509877 - 0.2675631950i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
1229 \( 1 \)
good2 \( 1 + (0.951 - 0.307i)T \)
3 \( 1 + (0.957 + 0.287i)T \)
7 \( 1 + (-0.998 + 0.0562i)T \)
11 \( 1 + (-0.0255 + 0.999i)T \)
13 \( 1 + (-0.730 - 0.683i)T \)
17 \( 1 + (0.995 + 0.0970i)T \)
19 \( 1 + (0.494 + 0.869i)T \)
23 \( 1 + (0.982 + 0.188i)T \)
29 \( 1 + (-0.994 - 0.102i)T \)
31 \( 1 + (-0.345 - 0.938i)T \)
37 \( 1 + (0.311 - 0.950i)T \)
41 \( 1 + (-0.962 - 0.272i)T \)
43 \( 1 + (0.985 + 0.168i)T \)
47 \( 1 + (0.879 - 0.476i)T \)
53 \( 1 + (-0.686 - 0.726i)T \)
59 \( 1 + (0.719 + 0.694i)T \)
61 \( 1 + (-0.107 - 0.994i)T \)
67 \( 1 + (0.0817 + 0.996i)T \)
71 \( 1 + (0.374 + 0.927i)T \)
73 \( 1 + (-0.416 - 0.909i)T \)
79 \( 1 + (-0.805 + 0.592i)T \)
83 \( 1 + (0.282 - 0.959i)T \)
89 \( 1 + (-0.448 + 0.893i)T \)
97 \( 1 + (0.737 - 0.675i)T \)
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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

−17.232209073094763028764256859206, −16.7117074908693424660055767236, −16.07919493675889245214804875384, −15.495679450741107317331281780194, −14.80588956593671351712136171645, −14.16422312980858653085660382597, −13.701403737385549048616616492907, −13.09581827429245113726913376796, −12.52743485500866012497548829665, −11.90814027560991613945006528171, −11.09761241546725191350433462052, −10.25689992832934659366210707224, −9.36471393676516918589368048729, −8.91980886656294742394189290267, −8.02187436344385346426500858338, −7.23304390150412635003336706366, −6.92488117069543021914419527554, −6.11505010489839212006196477993, −5.3345145192823139454375539464, −4.52175195953035582241342972729, −3.66324018214672768948458956144, −3.03204324440981214539514887192, −2.76819950841694061855511488938, −1.65710834836240350304291471442, −0.72406339251206225367253804232, 0.66836198474824174273873653725, 1.72787398113175154017832323401, 2.418730976766460651077772994, 3.08856321399733094812776222939, 3.69805387812551112020728111223, 4.251369191094393438820750172931, 5.32050396349607828808873666592, 5.62914693098739029991299997626, 6.79551864557915151883108678781, 7.436774153766195621267015515171, 7.791141633598993604947104637109, 9.11305007300384312477816877733, 9.70887982917985590645913159912, 10.07778079931047692577630795895, 10.73206574969925375205849262744, 11.808559710649527150021980170962, 12.49891151446893666778566342220, 12.92284321487329409104096539377, 13.39765559201487823380596916118, 14.47210139552954187532051512185, 14.631134431773519889637368507159, 15.35665627755354228968049973693, 15.88146759394907996528879688262, 16.604003387099573885396465260275, 17.25876951278299078133452651757

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