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Properties

Label 1-2415-2415.1214-r0-0-0
Degree $1$
Conductor $2415$
Sign $0.409 + 0.912i$
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  + (−0.888 + 0.458i)2-s + (0.580 − 0.814i)4-s + (−0.142 + 0.989i)8-s + (0.888 + 0.458i)11-s + (−0.654 − 0.755i)13-s + (−0.327 − 0.945i)16-s + (0.995 − 0.0950i)17-s + (0.995 + 0.0950i)19-s − 22-s + (0.928 + 0.371i)26-s + (−0.415 + 0.909i)29-s + (−0.928 + 0.371i)31-s + (0.723 + 0.690i)32-s + (−0.841 + 0.540i)34-s + (−0.235 + 0.971i)37-s + (−0.928 + 0.371i)38-s + ⋯
L(s)  = 1  + (−0.888 + 0.458i)2-s + (0.580 − 0.814i)4-s + (−0.142 + 0.989i)8-s + (0.888 + 0.458i)11-s + (−0.654 − 0.755i)13-s + (−0.327 − 0.945i)16-s + (0.995 − 0.0950i)17-s + (0.995 + 0.0950i)19-s − 22-s + (0.928 + 0.371i)26-s + (−0.415 + 0.909i)29-s + (−0.928 + 0.371i)31-s + (0.723 + 0.690i)32-s + (−0.841 + 0.540i)34-s + (−0.235 + 0.971i)37-s + (−0.928 + 0.371i)38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2415 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.409 + 0.912i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2415 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.409 + 0.912i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2415\)    =    \(3 \cdot 5 \cdot 7 \cdot 23\)
Sign: $0.409 + 0.912i$
Analytic conductor: \(11.2152\)
Root analytic conductor: \(11.2152\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2415} (1214, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2415,\ (0:\ ),\ 0.409 + 0.912i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9025884290 + 0.5845460043i\)
\(L(\frac12)\) \(\approx\) \(0.9025884290 + 0.5845460043i\)
\(L(1)\) \(\approx\) \(0.7572105520 + 0.1863616608i\)
\(L(1)\) \(\approx\) \(0.7572105520 + 0.1863616608i\)

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 \( 1 + (-0.888 + 0.458i)T \)
11 \( 1 + (0.888 + 0.458i)T \)
13 \( 1 + (-0.654 - 0.755i)T \)
17 \( 1 + (0.995 - 0.0950i)T \)
19 \( 1 + (0.995 + 0.0950i)T \)
29 \( 1 + (-0.415 + 0.909i)T \)
31 \( 1 + (-0.928 + 0.371i)T \)
37 \( 1 + (-0.235 + 0.971i)T \)
41 \( 1 + (-0.959 - 0.281i)T \)
43 \( 1 + (0.142 + 0.989i)T \)
47 \( 1 + (0.5 - 0.866i)T \)
53 \( 1 + (0.981 + 0.189i)T \)
59 \( 1 + (-0.327 + 0.945i)T \)
61 \( 1 + (0.786 + 0.618i)T \)
67 \( 1 + (-0.0475 + 0.998i)T \)
71 \( 1 + (-0.841 - 0.540i)T \)
73 \( 1 + (0.580 - 0.814i)T \)
79 \( 1 + (0.981 - 0.189i)T \)
83 \( 1 + (0.959 - 0.281i)T \)
89 \( 1 + (0.928 + 0.371i)T \)
97 \( 1 + (-0.959 - 0.281i)T \)
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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

−19.149793575455313692226453133742, −18.937016984506203093036532134149, −18.04238849953954170785088786025, −17.20160558950906543810723116127, −16.72870075578174334607583215163, −16.143180926729968690164216617952, −15.21087257746098304036444009091, −14.334178107589908723995666968373, −13.67575251431807556514167175638, −12.61117103524381156274631128710, −11.95571279888930252586873515850, −11.46291037275752731044799843086, −10.66442114790656604155973706327, −9.64153853589795172127356243601, −9.404067115038295799105787088880, −8.50279955041937656169807132411, −7.6213265295752306265620282337, −7.07199453673380324826988514892, −6.1684266609985424664864168349, −5.23353069001446112713871479920, −3.9587957480625870537318494391, −3.47292387573510340106098518246, −2.37313207381357372345796810605, −1.5825087221981204896097606227, −0.58706183324552612864844819613, 0.94567885503021412859266985092, 1.67218512114429393190523777619, 2.80645358860450113374643623568, 3.6761876673046806693994626717, 5.05146117155487234826469042415, 5.47239530383204057679895383588, 6.50524802315626347311938689098, 7.28912611012545601629708972263, 7.7306279223507546914340302942, 8.7439985124705230906268755093, 9.38167047878477555137518504363, 10.10006512234046698081287653693, 10.63780213366584986494002695768, 11.849919534889703257694903805820, 12.05517519094608698801594836138, 13.2620826110192239852977868335, 14.24059860246746244583807713210, 14.79717618520926755248414175156, 15.32811836107230912453928345179, 16.47645969530582197773298293836, 16.63563808598801831154859866231, 17.62916367043549476104621873532, 18.08962309278520900211909741938, 18.859217543493402766355922888429, 19.63411735775583483819472808591

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