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Properties

Label 2-2352-21.20-c1-0-66
Degree $2$
Conductor $2352$
Sign $0.0466 + 0.998i$
Analytic cond. $18.7808$
Root an. cond. $4.33368$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.07 − 1.36i)3-s + 2.57·5-s + (−0.703 − 2.91i)9-s − 1.65i·11-s − 5.71i·13-s + (2.76 − 3.50i)15-s + 7.58·17-s + 2.99i·19-s + 0.287i·23-s + 1.65·25-s + (−4.72 − 2.16i)27-s − 2.05i·29-s + 6.01i·31-s + (−2.25 − 1.77i)33-s + 1.75·37-s + ⋯
L(s)  = 1  + (0.618 − 0.785i)3-s + 1.15·5-s + (−0.234 − 0.972i)9-s − 0.498i·11-s − 1.58i·13-s + (0.713 − 0.906i)15-s + 1.83·17-s + 0.686i·19-s + 0.0600i·23-s + 0.330·25-s + (−0.908 − 0.417i)27-s − 0.382i·29-s + 1.08i·31-s + (−0.391 − 0.308i)33-s + 0.288·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0466 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0466 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2352\)    =    \(2^{4} \cdot 3 \cdot 7^{2}\)
Sign: $0.0466 + 0.998i$
Analytic conductor: \(18.7808\)
Root analytic conductor: \(4.33368\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2352} (881, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2352,\ (\ :1/2),\ 0.0466 + 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.836294133\)
\(L(\frac12)\) \(\approx\) \(2.836294133\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-1.07 + 1.36i)T \)
7 \( 1 \)
good5 \( 1 - 2.57T + 5T^{2} \)
11 \( 1 + 1.65iT - 11T^{2} \)
13 \( 1 + 5.71iT - 13T^{2} \)
17 \( 1 - 7.58T + 17T^{2} \)
19 \( 1 - 2.99iT - 19T^{2} \)
23 \( 1 - 0.287iT - 23T^{2} \)
29 \( 1 + 2.05iT - 29T^{2} \)
31 \( 1 - 6.01iT - 31T^{2} \)
37 \( 1 - 1.75T + 37T^{2} \)
41 \( 1 + 4.28T + 41T^{2} \)
43 \( 1 + 2.46T + 43T^{2} \)
47 \( 1 - 0.373T + 47T^{2} \)
53 \( 1 - 7.77iT - 53T^{2} \)
59 \( 1 + 9.79T + 59T^{2} \)
61 \( 1 + 1.02iT - 61T^{2} \)
67 \( 1 + 2.36T + 67T^{2} \)
71 \( 1 + 15.6iT - 71T^{2} \)
73 \( 1 + 3.81iT - 73T^{2} \)
79 \( 1 - 9.12T + 79T^{2} \)
83 \( 1 + 6.65T + 83T^{2} \)
89 \( 1 - 14.5T + 89T^{2} \)
97 \( 1 - 4.43iT - 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.708398687617814162718205836680, −7.920204076698156999617031733387, −7.52243463623006220253440940160, −6.25151301812506982782682498120, −5.87127748786512563014358572147, −5.11729626920694600623475985338, −3.43041712829528164027811250581, −3.02271671001888151531821364048, −1.81539351063710961057977383345, −0.920768721489632762991324679223, 1.60468629031090174308396807701, 2.38873523409369615945759108449, 3.43211684229824357808563849680, 4.38567712812480248537961378254, 5.15860231892789765729646578323, 5.92303586750226944874574624519, 6.87267183658679649347559437865, 7.70116680943400159569955539885, 8.620863012040669970927804327534, 9.396334798748645080365175183402

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