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Properties

Label 1-5160-5160.1493-r0-0-0
Degree $1$
Conductor $5160$
Sign $0.938 + 0.345i$
Analytic cond. $23.9629$
Root an. cond. $23.9629$
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  + (0.866 + 0.5i)7-s + (−0.222 + 0.974i)11-s + (−0.563 − 0.826i)13-s + (0.997 + 0.0747i)17-s + (−0.733 + 0.680i)19-s + (−0.294 − 0.955i)23-s + (−0.365 − 0.930i)29-s + (−0.988 − 0.149i)31-s + (0.866 − 0.5i)37-s + (−0.623 + 0.781i)41-s + (0.974 − 0.222i)47-s + (0.5 + 0.866i)49-s + (−0.563 + 0.826i)53-s + (0.900 − 0.433i)59-s + (0.988 − 0.149i)61-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)7-s + (−0.222 + 0.974i)11-s + (−0.563 − 0.826i)13-s + (0.997 + 0.0747i)17-s + (−0.733 + 0.680i)19-s + (−0.294 − 0.955i)23-s + (−0.365 − 0.930i)29-s + (−0.988 − 0.149i)31-s + (0.866 − 0.5i)37-s + (−0.623 + 0.781i)41-s + (0.974 − 0.222i)47-s + (0.5 + 0.866i)49-s + (−0.563 + 0.826i)53-s + (0.900 − 0.433i)59-s + (0.988 − 0.149i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5160 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.938 + 0.345i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5160 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.938 + 0.345i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5160\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 43\)
Sign: $0.938 + 0.345i$
Analytic conductor: \(23.9629\)
Root analytic conductor: \(23.9629\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5160} (1493, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5160,\ (0:\ ),\ 0.938 + 0.345i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.809650547 + 0.3222827185i\)
\(L(\frac12)\) \(\approx\) \(1.809650547 + 0.3222827185i\)
\(L(1)\) \(\approx\) \(1.139519131 + 0.08796089487i\)
\(L(1)\) \(\approx\) \(1.139519131 + 0.08796089487i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
43 \( 1 \)
good7 \( 1 + (0.866 + 0.5i)T \)
11 \( 1 + (-0.222 + 0.974i)T \)
13 \( 1 + (-0.563 - 0.826i)T \)
17 \( 1 + (0.997 + 0.0747i)T \)
19 \( 1 + (-0.733 + 0.680i)T \)
23 \( 1 + (-0.294 - 0.955i)T \)
29 \( 1 + (-0.365 - 0.930i)T \)
31 \( 1 + (-0.988 - 0.149i)T \)
37 \( 1 + (0.866 - 0.5i)T \)
41 \( 1 + (-0.623 + 0.781i)T \)
47 \( 1 + (0.974 - 0.222i)T \)
53 \( 1 + (-0.563 + 0.826i)T \)
59 \( 1 + (0.900 - 0.433i)T \)
61 \( 1 + (0.988 - 0.149i)T \)
67 \( 1 + (-0.680 - 0.733i)T \)
71 \( 1 + (-0.955 - 0.294i)T \)
73 \( 1 + (0.563 + 0.826i)T \)
79 \( 1 + (0.5 - 0.866i)T \)
83 \( 1 + (0.930 + 0.365i)T \)
89 \( 1 + (0.365 - 0.930i)T \)
97 \( 1 + (0.974 + 0.222i)T \)
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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

−17.913188354737976077078362452033, −17.2497997630033711899875160219, −16.611106025035577922779451456036, −16.188809081054077088671091220414, −15.15994490923960450652823121663, −14.564285251677897051028671105305, −14.03722247989716629357875518508, −13.400478026602511466931387154390, −12.65781611999469057013875335300, −11.71752787902624204157910128379, −11.342400405195544514166500179836, −10.61936864502272370254204808489, −9.95020034321442879419814133199, −9.02503717280787705904156153537, −8.53574546894006257708221144622, −7.59086991088345225063820111316, −7.24220114380487723927498075462, −6.27773860286791626782214836502, −5.40804978199514490117623018369, −4.91711197812469084399584378618, −3.96920798512619223447411616387, −3.38526713826534925700054542445, −2.32984175701926714443624023712, −1.58618284657143571610840973800, −0.671710247789438931441896128, 0.72255099559704927952427257188, 1.93988997239301239801077956051, 2.273451475664982329179755695138, 3.31943204228188716754895506127, 4.275300382917089143883974489680, 4.86274659176844161928800898287, 5.63854373515069707212254908055, 6.19516380850763665268504167868, 7.38492107202614004548310485934, 7.77848951441906658156760056849, 8.396888788653005910308994297115, 9.27978028889093561303966536830, 10.063179648747372136304785534705, 10.50496620872065261611952124574, 11.38415690998323396345727522762, 12.15710572949693861708282061970, 12.56499259698976458707045386523, 13.217856565779895456435016310440, 14.39475761327691762593139200551, 14.686854082070009077327521008, 15.18308476096731003818911484430, 15.9952669776473402526705409536, 16.93232488147903542747826969940, 17.25320532231922339678192987881, 18.220609608483539262578075405989

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