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Properties

Label 1-2415-2415.1307-r0-0-0
Degree $1$
Conductor $2415$
Sign $0.292 - 0.956i$
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  + (0.189 − 0.981i)2-s + (−0.928 − 0.371i)4-s + (−0.540 + 0.841i)8-s + (0.981 − 0.189i)11-s + (0.281 − 0.959i)13-s + (0.723 + 0.690i)16-s + (−0.618 + 0.786i)17-s + (0.786 − 0.618i)19-s − i·22-s + (−0.888 − 0.458i)26-s + (−0.142 + 0.989i)29-s + (0.888 − 0.458i)31-s + (0.814 − 0.580i)32-s + (0.654 + 0.755i)34-s + (0.0950 + 0.995i)37-s + (−0.458 − 0.888i)38-s + ⋯
L(s)  = 1  + (0.189 − 0.981i)2-s + (−0.928 − 0.371i)4-s + (−0.540 + 0.841i)8-s + (0.981 − 0.189i)11-s + (0.281 − 0.959i)13-s + (0.723 + 0.690i)16-s + (−0.618 + 0.786i)17-s + (0.786 − 0.618i)19-s − i·22-s + (−0.888 − 0.458i)26-s + (−0.142 + 0.989i)29-s + (0.888 − 0.458i)31-s + (0.814 − 0.580i)32-s + (0.654 + 0.755i)34-s + (0.0950 + 0.995i)37-s + (−0.458 − 0.888i)38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2415 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.292 - 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2415 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.292 - 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2415\)    =    \(3 \cdot 5 \cdot 7 \cdot 23\)
Sign: $0.292 - 0.956i$
Analytic conductor: \(11.2152\)
Root analytic conductor: \(11.2152\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2415} (1307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2415,\ (0:\ ),\ 0.292 - 0.956i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.408529587 - 1.041601913i\)
\(L(\frac12)\) \(\approx\) \(1.408529587 - 1.041601913i\)
\(L(1)\) \(\approx\) \(1.016696587 - 0.5600203873i\)
\(L(1)\) \(\approx\) \(1.016696587 - 0.5600203873i\)

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 \( 1 + (0.189 - 0.981i)T \)
11 \( 1 + (0.981 - 0.189i)T \)
13 \( 1 + (0.281 - 0.959i)T \)
17 \( 1 + (-0.618 + 0.786i)T \)
19 \( 1 + (0.786 - 0.618i)T \)
29 \( 1 + (-0.142 + 0.989i)T \)
31 \( 1 + (0.888 - 0.458i)T \)
37 \( 1 + (0.0950 + 0.995i)T \)
41 \( 1 + (0.415 + 0.909i)T \)
43 \( 1 + (0.540 + 0.841i)T \)
47 \( 1 + (0.866 - 0.5i)T \)
53 \( 1 + (-0.971 - 0.235i)T \)
59 \( 1 + (-0.723 + 0.690i)T \)
61 \( 1 + (0.0475 - 0.998i)T \)
67 \( 1 + (-0.945 + 0.327i)T \)
71 \( 1 + (0.654 - 0.755i)T \)
73 \( 1 + (-0.371 + 0.928i)T \)
79 \( 1 + (0.235 + 0.971i)T \)
83 \( 1 + (0.909 + 0.415i)T \)
89 \( 1 + (-0.888 - 0.458i)T \)
97 \( 1 + (-0.909 + 0.415i)T \)
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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

−19.41474861357365859362722242234, −18.91568874045648667591770801902, −18.042655153145381286122948360491, −17.43273626897676874629226272777, −16.76975779305213042016955904481, −16.017090718582785031450327755725, −15.55253421971333667405255309841, −14.58498024874288672831340020190, −13.91610621817257174798288340884, −13.65620212563039206054637399195, −12.40272474328628321019366945724, −11.95782688335618622360270296303, −11.06265182114888597864073216053, −9.8898263805905126341133277487, −9.20341900329059399936097596024, −8.77041290576378495255877652556, −7.6487575557841294954667434743, −7.112899667383567026806011494285, −6.303695055952118588794249106664, −5.701517810236018974750836883866, −4.58479699509561644240752382186, −4.13273970870977113307336343662, −3.20745445661633718241197100618, −1.97075779216348271232090880873, −0.77259440045641397498313833609, 0.85033360529772842209641184004, 1.5408756753902031195042927512, 2.713462627044326846719226375441, 3.33969426815485053674751668080, 4.209330357996665563848418437325, 4.94409360581339020729034419002, 5.89184202454397390304209878934, 6.56849544473019968840416389305, 7.80939836558167382083799870721, 8.55803648282274885783861599590, 9.27839649457471417260916339249, 9.96278031269627496202481451157, 10.85150281371599384193277498926, 11.325939204745822242489265433612, 12.141953985565212526688719276945, 12.85062395773441568578293339057, 13.49625295716417261776611521863, 14.19017587666551391882527550633, 14.99444631282236566777026862056, 15.60666177074726160394391234259, 16.74420052740428425696088433028, 17.4656698452190658564273724578, 18.02784492269843233043465771039, 18.7812625684489831239502378225, 19.67560429689231588861611012204

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