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Properties

Label 1-2415-2415.1664-r0-0-0
Degree $1$
Conductor $2415$
Sign $-0.995 + 0.0898i$
Analytic cond. $11.2152$
Root an. cond. $11.2152$
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  + (0.235 + 0.971i)2-s + (−0.888 + 0.458i)4-s + (−0.654 − 0.755i)8-s + (−0.235 + 0.971i)11-s + (0.415 − 0.909i)13-s + (0.580 − 0.814i)16-s + (−0.0475 − 0.998i)17-s + (−0.0475 + 0.998i)19-s − 22-s + (0.981 + 0.189i)26-s + (−0.841 + 0.540i)29-s + (−0.981 + 0.189i)31-s + (0.928 + 0.371i)32-s + (0.959 − 0.281i)34-s + (0.786 − 0.618i)37-s + (−0.981 + 0.189i)38-s + ⋯
L(s)  = 1  + (0.235 + 0.971i)2-s + (−0.888 + 0.458i)4-s + (−0.654 − 0.755i)8-s + (−0.235 + 0.971i)11-s + (0.415 − 0.909i)13-s + (0.580 − 0.814i)16-s + (−0.0475 − 0.998i)17-s + (−0.0475 + 0.998i)19-s − 22-s + (0.981 + 0.189i)26-s + (−0.841 + 0.540i)29-s + (−0.981 + 0.189i)31-s + (0.928 + 0.371i)32-s + (0.959 − 0.281i)34-s + (0.786 − 0.618i)37-s + (−0.981 + 0.189i)38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2415 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.995 + 0.0898i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2415 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.995 + 0.0898i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2415\)    =    \(3 \cdot 5 \cdot 7 \cdot 23\)
Sign: $-0.995 + 0.0898i$
Analytic conductor: \(11.2152\)
Root analytic conductor: \(11.2152\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2415} (1664, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2415,\ (0:\ ),\ -0.995 + 0.0898i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.04626137238 + 1.027633385i\)
\(L(\frac12)\) \(\approx\) \(0.04626137238 + 1.027633385i\)
\(L(1)\) \(\approx\) \(0.7775110685 + 0.5640283419i\)
\(L(1)\) \(\approx\) \(0.7775110685 + 0.5640283419i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (0.235 + 0.971i)T \)
11 \( 1 + (-0.235 + 0.971i)T \)
13 \( 1 + (0.415 - 0.909i)T \)
17 \( 1 + (-0.0475 - 0.998i)T \)
19 \( 1 + (-0.0475 + 0.998i)T \)
29 \( 1 + (-0.841 + 0.540i)T \)
31 \( 1 + (-0.981 + 0.189i)T \)
37 \( 1 + (0.786 - 0.618i)T \)
41 \( 1 + (-0.142 + 0.989i)T \)
43 \( 1 + (0.654 - 0.755i)T \)
47 \( 1 + (0.5 + 0.866i)T \)
53 \( 1 + (-0.995 - 0.0950i)T \)
59 \( 1 + (0.580 + 0.814i)T \)
61 \( 1 + (0.327 - 0.945i)T \)
67 \( 1 + (-0.723 + 0.690i)T \)
71 \( 1 + (0.959 + 0.281i)T \)
73 \( 1 + (-0.888 + 0.458i)T \)
79 \( 1 + (-0.995 + 0.0950i)T \)
83 \( 1 + (0.142 + 0.989i)T \)
89 \( 1 + (0.981 + 0.189i)T \)
97 \( 1 + (-0.142 + 0.989i)T \)
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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

−19.09346139587919651189850987292, −18.812318288990294479017213628007, −17.93803599326262724730189224638, −17.14563210031006941250996667269, −16.43028774029123669560168080824, −15.46949932557900065082283327271, −14.718581561169045078533384519452, −13.960090416728335934642337865999, −13.28929089558605857499410201718, −12.80853945432777314015485480311, −11.769546574900357958805747891646, −11.18611290534963539057785503395, −10.719274596513418382101664317882, −9.74553485093497129227923840287, −8.970587811554776709019244047525, −8.48720246838284469057901796052, −7.437063947012654421176816108293, −6.24915742891002562988606740244, −5.7262381235703936942214536869, −4.70043378539644636673301558344, −3.94882017681231870000176260429, −3.25500699171904762988656514008, −2.27622676828786417054132871528, −1.505639132414355664281803168639, −0.34060585079109575297435459777, 1.10052061524409767841890853940, 2.42714619064129530687235067188, 3.432514630453084164911016134752, 4.187059354488060352256835935551, 5.12776831552806714700009161219, 5.654530961570293727599920466046, 6.54048933671896049118587688227, 7.50469292845439327863719698572, 7.75076106228391159815920108017, 8.82733562305616147167359165717, 9.50582986344822306748244685984, 10.23813972181510445102068910639, 11.19006925819134930683578995978, 12.2560275624072749338608616887, 12.80756788902150875890808300049, 13.39781767616200785843122773862, 14.48549217455763657008729329831, 14.705594641823200036744376605417, 15.75262611265851903708963725195, 16.09200339378093561032711416046, 16.98545733461617066705966317038, 17.69933345253869629669511849678, 18.30455621636210548418503879574, 18.79199795780336720326263897343, 20.089428456326811737067395561932

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