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Properties

Label 1-2415-2415.1844-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: $\chi_{2415} (1844, \cdot )$
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

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

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