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Properties

Label 1-612-612.259-r1-0-0
Degree $1$
Conductor $612$
Sign $-0.883 - 0.469i$
Analytic cond. $65.7685$
Root an. cond. $65.7685$
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)5-s + (−0.866 + 0.5i)7-s + (−0.866 + 0.5i)11-s + (−0.5 + 0.866i)13-s + 19-s + (0.866 + 0.5i)23-s + (0.5 + 0.866i)25-s + (−0.866 + 0.5i)29-s + (−0.866 − 0.5i)31-s − 35-s − i·37-s + (−0.866 − 0.5i)41-s + (−0.5 − 0.866i)43-s + (0.5 + 0.866i)47-s + (0.5 − 0.866i)49-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)5-s + (−0.866 + 0.5i)7-s + (−0.866 + 0.5i)11-s + (−0.5 + 0.866i)13-s + 19-s + (0.866 + 0.5i)23-s + (0.5 + 0.866i)25-s + (−0.866 + 0.5i)29-s + (−0.866 − 0.5i)31-s − 35-s − i·37-s + (−0.866 − 0.5i)41-s + (−0.5 − 0.866i)43-s + (0.5 + 0.866i)47-s + (0.5 − 0.866i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 612 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.883 - 0.469i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 612 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.883 - 0.469i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(612\)    =    \(2^{2} \cdot 3^{2} \cdot 17\)
Sign: $-0.883 - 0.469i$
Analytic conductor: \(65.7685\)
Root analytic conductor: \(65.7685\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{612} (259, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 612,\ (1:\ ),\ -0.883 - 0.469i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.09540903430 + 0.3829227012i\)
\(L(\frac12)\) \(\approx\) \(-0.09540903430 + 0.3829227012i\)
\(L(1)\) \(\approx\) \(0.8642033524 + 0.2486131522i\)
\(L(1)\) \(\approx\) \(0.8642033524 + 0.2486131522i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
17 \( 1 \)
good5 \( 1 + (0.866 + 0.5i)T \)
7 \( 1 + (-0.866 + 0.5i)T \)
11 \( 1 + (-0.866 + 0.5i)T \)
13 \( 1 + (-0.5 + 0.866i)T \)
19 \( 1 + T \)
23 \( 1 + (0.866 + 0.5i)T \)
29 \( 1 + (-0.866 + 0.5i)T \)
31 \( 1 + (-0.866 - 0.5i)T \)
37 \( 1 - iT \)
41 \( 1 + (-0.866 - 0.5i)T \)
43 \( 1 + (-0.5 - 0.866i)T \)
47 \( 1 + (0.5 + 0.866i)T \)
53 \( 1 - T \)
59 \( 1 + (-0.5 + 0.866i)T \)
61 \( 1 + (0.866 - 0.5i)T \)
67 \( 1 + (0.5 - 0.866i)T \)
71 \( 1 - iT \)
73 \( 1 - iT \)
79 \( 1 + (-0.866 + 0.5i)T \)
83 \( 1 + (-0.5 - 0.866i)T \)
89 \( 1 + T \)
97 \( 1 + (-0.866 + 0.5i)T \)
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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

−22.27722225008360645258683555530, −21.63021481773371080194382256049, −20.46740690904437191251664841560, −20.21718313688049738329146104510, −18.96663154430347965171437668336, −18.226595665923756420735575787730, −17.25457737152801886140641507301, −16.585270894133360241028650610828, −15.8400900865394222626826683677, −14.79300396229439119284800818296, −13.66534821927820368448035720636, −13.13575108920864980301880800235, −12.50210778095989812373848261177, −11.16030316689218070228092404835, −10.160549009182770409788380167498, −9.6879310338161606527267113782, −8.61109078303353629791404001071, −7.597392565034497180064213514474, −6.5857262917475046974087130556, −5.55595375583413445180849417050, −4.92608250293959050017162146290, −3.40162806959528072872605841623, −2.62954488011918370167237005876, −1.170805268832044619818929205325, −0.09346994510256607597638448387, 1.74785483355286004305356905976, 2.615378352226111321764645484822, 3.55374823670906440066982328067, 5.10431585826206078664828861690, 5.73068881491345424481963492641, 6.88297043560690482251916436541, 7.454552534922015809391116455233, 9.09986469493126946795685808900, 9.51208772997280266348464180285, 10.40791477533569089930881052059, 11.38140132109734101936181381347, 12.498798078492825558637046209197, 13.17973210964391064811974530881, 14.044143204499813208667675647615, 14.949908264912354986277897778543, 15.77207083416223112898620457668, 16.69301683282679888678570886260, 17.5389331389498053093660420586, 18.54281185515901890473426822992, 18.872124305454373339684334369422, 20.06818302492790155698797181235, 20.93155662010394644613006003158, 21.82657516027108233285983256504, 22.30536174491850706658187718268, 23.20378185194730403310954907503

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