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Properties

Label 1-2415-2415.404-r0-0-0
Degree $1$
Conductor $2415$
Sign $-0.633 - 0.773i$
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  + (0.928 − 0.371i)2-s + (0.723 − 0.690i)4-s + (0.415 − 0.909i)8-s + (−0.928 − 0.371i)11-s + (0.841 − 0.540i)13-s + (0.0475 − 0.998i)16-s + (−0.235 + 0.971i)17-s + (−0.235 − 0.971i)19-s − 22-s + (0.580 − 0.814i)26-s + (0.959 − 0.281i)29-s + (−0.580 − 0.814i)31-s + (−0.327 − 0.945i)32-s + (0.142 + 0.989i)34-s + (−0.981 − 0.189i)37-s + (−0.580 − 0.814i)38-s + ⋯
L(s)  = 1  + (0.928 − 0.371i)2-s + (0.723 − 0.690i)4-s + (0.415 − 0.909i)8-s + (−0.928 − 0.371i)11-s + (0.841 − 0.540i)13-s + (0.0475 − 0.998i)16-s + (−0.235 + 0.971i)17-s + (−0.235 − 0.971i)19-s − 22-s + (0.580 − 0.814i)26-s + (0.959 − 0.281i)29-s + (−0.580 − 0.814i)31-s + (−0.327 − 0.945i)32-s + (0.142 + 0.989i)34-s + (−0.981 − 0.189i)37-s + (−0.580 − 0.814i)38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2415 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.633 - 0.773i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2415 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.633 - 0.773i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2415\)    =    \(3 \cdot 5 \cdot 7 \cdot 23\)
Sign: $-0.633 - 0.773i$
Analytic conductor: \(11.2152\)
Root analytic conductor: \(11.2152\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2415} (404, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2415,\ (0:\ ),\ -0.633 - 0.773i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.054101706 - 2.225684650i\)
\(L(\frac12)\) \(\approx\) \(1.054101706 - 2.225684650i\)
\(L(1)\) \(\approx\) \(1.505137141 - 0.7722126133i\)
\(L(1)\) \(\approx\) \(1.505137141 - 0.7722126133i\)

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 \( 1 + (0.928 - 0.371i)T \)
11 \( 1 + (-0.928 - 0.371i)T \)
13 \( 1 + (0.841 - 0.540i)T \)
17 \( 1 + (-0.235 + 0.971i)T \)
19 \( 1 + (-0.235 - 0.971i)T \)
29 \( 1 + (0.959 - 0.281i)T \)
31 \( 1 + (-0.580 - 0.814i)T \)
37 \( 1 + (-0.981 - 0.189i)T \)
41 \( 1 + (-0.654 - 0.755i)T \)
43 \( 1 + (-0.415 - 0.909i)T \)
47 \( 1 + (0.5 + 0.866i)T \)
53 \( 1 + (-0.888 + 0.458i)T \)
59 \( 1 + (0.0475 + 0.998i)T \)
61 \( 1 + (0.995 - 0.0950i)T \)
67 \( 1 + (0.786 + 0.618i)T \)
71 \( 1 + (0.142 - 0.989i)T \)
73 \( 1 + (0.723 - 0.690i)T \)
79 \( 1 + (-0.888 - 0.458i)T \)
83 \( 1 + (0.654 - 0.755i)T \)
89 \( 1 + (0.580 - 0.814i)T \)
97 \( 1 + (-0.654 - 0.755i)T \)
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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

−20.18350181572311326527179461357, −19.02518835880214049334012166943, −18.28185667399114242784826460552, −17.62204425908314204538297259513, −16.6475967912506515999861399956, −16.01398844691037172604719001112, −15.62860352606669640911730206365, −14.650268121890835163951336321206, −14.040066943958947051080563380304, −13.4002549993413736649526085140, −12.670557492726207611997604963854, −12.008004405270848802870390765549, −11.216830302678129818906173207836, −10.52889659963099422011548013708, −9.61656054661433926262528732867, −8.440772566601896231363422232682, −8.03112921388855585760563510583, −6.88523616356189190139678968387, −6.5740914671254734606551053274, −5.38086202519270874743931420707, −4.98466750736099328595056278341, −3.98637960189249902694759842220, −3.24530309953017256555317721126, −2.35930169945004135557700818496, −1.45202991567057970984327135149, 0.52822471387604365858304560100, 1.7132993402573769994373595100, 2.572505243284561906233752804640, 3.36878910653390352933296242960, 4.1264790664532334412927344113, 5.038964951493209501666496325609, 5.73824895861007089916847748346, 6.40492522833741787421329563628, 7.30084866446871218008786082372, 8.22512188828844628605332670841, 8.99274852579523129441616143230, 10.18995100016789123244814750031, 10.67637227328995836828009648653, 11.23659241941601764739532839529, 12.18778371476563869698204079264, 12.9115571483104189078353952084, 13.43707768831222256283155664510, 14.0291028588037793737920273768, 15.08711326081428982345629809456, 15.53597102607554087451617068281, 16.08142544947983979057152613800, 17.130869680552126502242445739563, 17.89644332867276605415402075129, 18.842571301318624399200680101466, 19.27731056995007351373668735921

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