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Properties

Label 1-2415-2415.2183-r0-0-0
Degree $1$
Conductor $2415$
Sign $0.888 - 0.458i$
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  + (−0.909 + 0.415i)2-s + (0.654 − 0.755i)4-s + (−0.281 + 0.959i)8-s + (0.415 − 0.909i)11-s + (0.989 + 0.142i)13-s + (−0.142 − 0.989i)16-s + (0.755 − 0.654i)17-s + (0.654 − 0.755i)19-s + i·22-s + (−0.959 + 0.281i)26-s + (−0.654 − 0.755i)29-s + (0.959 + 0.281i)31-s + (0.540 + 0.841i)32-s + (−0.415 + 0.909i)34-s + (0.540 + 0.841i)37-s + (−0.281 + 0.959i)38-s + ⋯
L(s)  = 1  + (−0.909 + 0.415i)2-s + (0.654 − 0.755i)4-s + (−0.281 + 0.959i)8-s + (0.415 − 0.909i)11-s + (0.989 + 0.142i)13-s + (−0.142 − 0.989i)16-s + (0.755 − 0.654i)17-s + (0.654 − 0.755i)19-s + i·22-s + (−0.959 + 0.281i)26-s + (−0.654 − 0.755i)29-s + (0.959 + 0.281i)31-s + (0.540 + 0.841i)32-s + (−0.415 + 0.909i)34-s + (0.540 + 0.841i)37-s + (−0.281 + 0.959i)38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2415 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.888 - 0.458i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2415 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.888 - 0.458i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2415\)    =    \(3 \cdot 5 \cdot 7 \cdot 23\)
Sign: $0.888 - 0.458i$
Analytic conductor: \(11.2152\)
Root analytic conductor: \(11.2152\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2415} (2183, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2415,\ (0:\ ),\ 0.888 - 0.458i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.230312002 - 0.2986785178i\)
\(L(\frac12)\) \(\approx\) \(1.230312002 - 0.2986785178i\)
\(L(1)\) \(\approx\) \(0.8486059764 + 0.001761907162i\)
\(L(1)\) \(\approx\) \(0.8486059764 + 0.001761907162i\)

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 \( 1 + (-0.909 + 0.415i)T \)
11 \( 1 + (0.415 - 0.909i)T \)
13 \( 1 + (0.989 + 0.142i)T \)
17 \( 1 + (0.755 - 0.654i)T \)
19 \( 1 + (0.654 - 0.755i)T \)
29 \( 1 + (-0.654 - 0.755i)T \)
31 \( 1 + (0.959 + 0.281i)T \)
37 \( 1 + (0.540 + 0.841i)T \)
41 \( 1 + (0.841 + 0.540i)T \)
43 \( 1 + (0.281 + 0.959i)T \)
47 \( 1 - iT \)
53 \( 1 + (0.989 - 0.142i)T \)
59 \( 1 + (0.142 - 0.989i)T \)
61 \( 1 + (-0.959 - 0.281i)T \)
67 \( 1 + (-0.909 + 0.415i)T \)
71 \( 1 + (-0.415 - 0.909i)T \)
73 \( 1 + (0.755 + 0.654i)T \)
79 \( 1 + (-0.142 + 0.989i)T \)
83 \( 1 + (-0.540 - 0.841i)T \)
89 \( 1 + (-0.959 + 0.281i)T \)
97 \( 1 + (0.540 - 0.841i)T \)
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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

−19.56337085804470282480984756835, −18.8117717676791638077318416225, −18.19064547490547036289022163563, −17.58756909542132604608449091476, −16.81331380947447406890451596973, −16.22475013438095560592015138684, −15.44239135439547098530403389508, −14.700448427720392078690174901113, −13.78406027832914662742282121801, −12.799136846364022497733789816509, −12.26858889343897351703084003014, −11.57700166365510646531619986593, −10.67135098839156878771756260693, −10.17841574276995365121286121922, −9.31685751125195393055579419059, −8.74727707479564342703678358065, −7.76285100911535665152397960058, −7.35064930626697875744418624775, −6.28024961314274072185287737248, −5.62644584375132591670360045953, −4.20460158651380983723519845314, −3.648762796383611786996237954218, −2.672246389022918626910328774063, −1.647896756676870181870788783721, −1.03295394634791138681782556326, 0.72120524583602054968998791520, 1.35094816850712041712829138957, 2.63159067718355031090972843433, 3.36828784978290319639345799342, 4.57614849405718162155416343987, 5.58891876949126722309109068000, 6.176680921105509715219687335292, 6.9330455133746247847325571735, 7.8307340748058591739579482749, 8.42370317703883528593861049379, 9.23344041301764875942418496913, 9.77659481500498628225586445996, 10.71601347733736962293281261944, 11.48289982608896203697999631547, 11.79491960340036307950237771886, 13.20870883114169238486112529743, 13.84963728797971518441022423442, 14.52059624439127614367014396395, 15.45735538820224585717382398873, 16.01149778230000058257243152063, 16.6574347667854551480617877596, 17.24712650184993636898916946296, 18.32641510204410378655092458852, 18.45317815832684479924783989800, 19.43729794627071461116058074107

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