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Properties

Label 1-2415-2415.509-r0-0-0
Degree $1$
Conductor $2415$
Sign $-0.986 + 0.162i$
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  + (0.723 + 0.690i)2-s + (0.0475 + 0.998i)4-s + (−0.654 + 0.755i)8-s + (−0.723 + 0.690i)11-s + (0.415 + 0.909i)13-s + (−0.995 + 0.0950i)16-s + (0.888 − 0.458i)17-s + (0.888 + 0.458i)19-s − 22-s + (−0.327 + 0.945i)26-s + (−0.841 − 0.540i)29-s + (0.327 + 0.945i)31-s + (−0.786 − 0.618i)32-s + (0.959 + 0.281i)34-s + (−0.928 + 0.371i)37-s + (0.327 + 0.945i)38-s + ⋯
L(s)  = 1  + (0.723 + 0.690i)2-s + (0.0475 + 0.998i)4-s + (−0.654 + 0.755i)8-s + (−0.723 + 0.690i)11-s + (0.415 + 0.909i)13-s + (−0.995 + 0.0950i)16-s + (0.888 − 0.458i)17-s + (0.888 + 0.458i)19-s − 22-s + (−0.327 + 0.945i)26-s + (−0.841 − 0.540i)29-s + (0.327 + 0.945i)31-s + (−0.786 − 0.618i)32-s + (0.959 + 0.281i)34-s + (−0.928 + 0.371i)37-s + (0.327 + 0.945i)38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2415 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.986 + 0.162i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2415 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.986 + 0.162i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2415\)    =    \(3 \cdot 5 \cdot 7 \cdot 23\)
Sign: $-0.986 + 0.162i$
Analytic conductor: \(11.2152\)
Root analytic conductor: \(11.2152\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2415} (509, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2415,\ (0:\ ),\ -0.986 + 0.162i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1602872110 + 1.956008340i\)
\(L(\frac12)\) \(\approx\) \(0.1602872110 + 1.956008340i\)
\(L(1)\) \(\approx\) \(1.100668271 + 0.9037181676i\)
\(L(1)\) \(\approx\) \(1.100668271 + 0.9037181676i\)

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 \( 1 + (0.723 + 0.690i)T \)
11 \( 1 + (-0.723 + 0.690i)T \)
13 \( 1 + (0.415 + 0.909i)T \)
17 \( 1 + (0.888 - 0.458i)T \)
19 \( 1 + (0.888 + 0.458i)T \)
29 \( 1 + (-0.841 - 0.540i)T \)
31 \( 1 + (0.327 + 0.945i)T \)
37 \( 1 + (-0.928 + 0.371i)T \)
41 \( 1 + (-0.142 - 0.989i)T \)
43 \( 1 + (0.654 + 0.755i)T \)
47 \( 1 + (0.5 + 0.866i)T \)
53 \( 1 + (0.580 + 0.814i)T \)
59 \( 1 + (-0.995 - 0.0950i)T \)
61 \( 1 + (-0.981 - 0.189i)T \)
67 \( 1 + (-0.235 + 0.971i)T \)
71 \( 1 + (0.959 - 0.281i)T \)
73 \( 1 + (0.0475 + 0.998i)T \)
79 \( 1 + (0.580 - 0.814i)T \)
83 \( 1 + (0.142 - 0.989i)T \)
89 \( 1 + (-0.327 + 0.945i)T \)
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

−19.323179697984515467450985707743, −18.47156319104743218091658847349, −18.173314749525534693541540076254, −16.99915567007537401596350596874, −16.169320044531841647480697165647, −15.40154232507507580467985976248, −14.9235598243675465959158399333, −13.8456936788321558214164090109, −13.50761023283696575880456375156, −12.68273079823350792192679338861, −12.06951794583271302300816312128, −11.13867591447973590776574784660, −10.65892039634578330601188634869, −9.927548939212834610644035857945, −9.10042019654709892909435851914, −8.136664411753771822474619578620, −7.36135092899191080250984081529, −6.2388682591376814139806704556, −5.50834360668648493564206155042, −5.10607768883575452660026951762, −3.8338717051697654913009254346, −3.29091255481503982662989834086, −2.52880776930068980427817925702, −1.42917904611844476340464326725, −0.48920436396080056830632789829, 1.42456486494456695162318088329, 2.510140680893267784444512121791, 3.34707824947035433995498755544, 4.17115384995467630959294398843, 5.00034884693075650154449675423, 5.63267694933440472284885888153, 6.473563004637156677577717871170, 7.40929121486454911273810458039, 7.71820176743948040419862696423, 8.79785958782719028020970034505, 9.50447928654220854863630189476, 10.44808502544228962626530570994, 11.41255669835205224232247251328, 12.16587723760511718425046727817, 12.61318291997975076678117276891, 13.75015138683332202708982059605, 13.95603225589143838580344565991, 14.84965041062451469864652908960, 15.66842293788664656879432004077, 16.10201227003293759445422209980, 16.876006241500416655947368342501, 17.58241064823257152835364229180, 18.39428828747036750462030894519, 18.936532731483031350705693995470, 20.14607268906526656299720135391

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