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Properties

Label 1-2415-2415.719-r0-0-0
Degree $1$
Conductor $2415$
Sign $0.574 - 0.818i$
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  + (−0.327 + 0.945i)2-s + (−0.786 − 0.618i)4-s + (0.841 − 0.540i)8-s + (0.327 + 0.945i)11-s + (−0.959 + 0.281i)13-s + (0.235 + 0.971i)16-s + (−0.928 − 0.371i)17-s + (−0.928 + 0.371i)19-s − 22-s + (0.0475 − 0.998i)26-s + (0.142 + 0.989i)29-s + (−0.0475 − 0.998i)31-s + (−0.995 − 0.0950i)32-s + (0.654 − 0.755i)34-s + (−0.580 + 0.814i)37-s + (−0.0475 − 0.998i)38-s + ⋯
L(s)  = 1  + (−0.327 + 0.945i)2-s + (−0.786 − 0.618i)4-s + (0.841 − 0.540i)8-s + (0.327 + 0.945i)11-s + (−0.959 + 0.281i)13-s + (0.235 + 0.971i)16-s + (−0.928 − 0.371i)17-s + (−0.928 + 0.371i)19-s − 22-s + (0.0475 − 0.998i)26-s + (0.142 + 0.989i)29-s + (−0.0475 − 0.998i)31-s + (−0.995 − 0.0950i)32-s + (0.654 − 0.755i)34-s + (−0.580 + 0.814i)37-s + (−0.0475 − 0.998i)38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2415 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.574 - 0.818i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2415 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.574 - 0.818i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2415\)    =    \(3 \cdot 5 \cdot 7 \cdot 23\)
Sign: $0.574 - 0.818i$
Analytic conductor: \(11.2152\)
Root analytic conductor: \(11.2152\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2415} (719, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2415,\ (0:\ ),\ 0.574 - 0.818i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3215572643 - 0.1670975432i\)
\(L(\frac12)\) \(\approx\) \(0.3215572643 - 0.1670975432i\)
\(L(1)\) \(\approx\) \(0.6126019319 + 0.2692863449i\)
\(L(1)\) \(\approx\) \(0.6126019319 + 0.2692863449i\)

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 \( 1 + (-0.327 + 0.945i)T \)
11 \( 1 + (0.327 + 0.945i)T \)
13 \( 1 + (-0.959 + 0.281i)T \)
17 \( 1 + (-0.928 - 0.371i)T \)
19 \( 1 + (-0.928 + 0.371i)T \)
29 \( 1 + (0.142 + 0.989i)T \)
31 \( 1 + (-0.0475 - 0.998i)T \)
37 \( 1 + (-0.580 + 0.814i)T \)
41 \( 1 + (0.415 - 0.909i)T \)
43 \( 1 + (-0.841 - 0.540i)T \)
47 \( 1 + (0.5 + 0.866i)T \)
53 \( 1 + (0.723 - 0.690i)T \)
59 \( 1 + (0.235 - 0.971i)T \)
61 \( 1 + (0.888 + 0.458i)T \)
67 \( 1 + (-0.981 + 0.189i)T \)
71 \( 1 + (0.654 + 0.755i)T \)
73 \( 1 + (-0.786 - 0.618i)T \)
79 \( 1 + (0.723 + 0.690i)T \)
83 \( 1 + (-0.415 - 0.909i)T \)
89 \( 1 + (0.0475 - 0.998i)T \)
97 \( 1 + (0.415 - 0.909i)T \)
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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

−19.50981257680475499399985323105, −19.269178523402854667446932729268, −18.236229674019921036492870449204, −17.60116380363257533079902646609, −16.98822757826570006725207519437, −16.31822836050959765305595624453, −15.263180901755088558966802097152, −14.5224631976708066382806400623, −13.638677403323238358132779168170, −13.112365905858563611534984762627, −12.287772817606590313085807555846, −11.64854010078035615759470129321, −10.84815183988653935316736705191, −10.35370204241781907366759279515, −9.42191241335392210149881282897, −8.733914750373731973075952101555, −8.16918502212626965670839558014, −7.188058417169995662819225438179, −6.29036393549619848008698443004, −5.240930738090823375941652093082, −4.4167332045577618630862199020, −3.68684343672239059916473276447, −2.71394066665163437136379599779, −2.0913760847755215417440708953, −0.93847823005669353872333109337, 0.15390864749194707780505948140, 1.61282992865622805431514747526, 2.39058305368980992440468412462, 3.863178701186791400294117341177, 4.56859506189988030894088962024, 5.19327673237914306142945045064, 6.221902029258775361791292632086, 6.97108504652894396609446333480, 7.38163667522495468365285778931, 8.445271696358038351147628304282, 9.035686477078075390069491228328, 9.83785746569783261073182171796, 10.38571012239230497369820636309, 11.4128447116218592553393831072, 12.36377935599388359731514510914, 13.01150175418496934406811664662, 13.88427810733695090373715137853, 14.64367912804275784856357922845, 15.089955777397952405142247695479, 15.831905957796149369562710793181, 16.6681614760608079311013930877, 17.29673870066412732203205136921, 17.73890133328671457871006489867, 18.64712666159674028453640471169, 19.26142607422617648439540184686

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