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Properties

Label 2-2340-15.8-c1-0-21
Degree $2$
Conductor $2340$
Sign $0.242 + 0.970i$
Analytic cond. $18.6849$
Root an. cond. $4.32261$
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  + (1.80 − 1.31i)5-s + (−0.602 + 0.602i)7-s − 1.84i·11-s + (−0.707 − 0.707i)13-s + (3.70 + 3.70i)17-s − 7.06i·19-s + (−0.688 + 0.688i)23-s + (1.53 − 4.75i)25-s − 5.94·29-s − 0.288·31-s + (−0.296 + 1.88i)35-s + (5.56 − 5.56i)37-s − 5.63i·41-s + (5.76 + 5.76i)43-s + (−4.61 − 4.61i)47-s + ⋯
L(s)  = 1  + (0.808 − 0.588i)5-s + (−0.227 + 0.227i)7-s − 0.556i·11-s + (−0.196 − 0.196i)13-s + (0.898 + 0.898i)17-s − 1.62i·19-s + (−0.143 + 0.143i)23-s + (0.307 − 0.951i)25-s − 1.10·29-s − 0.0517·31-s + (−0.0501 + 0.318i)35-s + (0.914 − 0.914i)37-s − 0.879i·41-s + (0.878 + 0.878i)43-s + (−0.672 − 0.672i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.242 + 0.970i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.242 + 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2340\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 13\)
Sign: $0.242 + 0.970i$
Analytic conductor: \(18.6849\)
Root analytic conductor: \(4.32261\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2340} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2340,\ (\ :1/2),\ 0.242 + 0.970i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.851920373\)
\(L(\frac12)\) \(\approx\) \(1.851920373\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 + (-1.80 + 1.31i)T \)
13 \( 1 + (0.707 + 0.707i)T \)
good7 \( 1 + (0.602 - 0.602i)T - 7iT^{2} \)
11 \( 1 + 1.84iT - 11T^{2} \)
17 \( 1 + (-3.70 - 3.70i)T + 17iT^{2} \)
19 \( 1 + 7.06iT - 19T^{2} \)
23 \( 1 + (0.688 - 0.688i)T - 23iT^{2} \)
29 \( 1 + 5.94T + 29T^{2} \)
31 \( 1 + 0.288T + 31T^{2} \)
37 \( 1 + (-5.56 + 5.56i)T - 37iT^{2} \)
41 \( 1 + 5.63iT - 41T^{2} \)
43 \( 1 + (-5.76 - 5.76i)T + 43iT^{2} \)
47 \( 1 + (4.61 + 4.61i)T + 47iT^{2} \)
53 \( 1 + (-7.46 + 7.46i)T - 53iT^{2} \)
59 \( 1 + 9.07T + 59T^{2} \)
61 \( 1 - 9.04T + 61T^{2} \)
67 \( 1 + (-0.873 + 0.873i)T - 67iT^{2} \)
71 \( 1 + 6.96iT - 71T^{2} \)
73 \( 1 + (4.12 + 4.12i)T + 73iT^{2} \)
79 \( 1 + 6.25iT - 79T^{2} \)
83 \( 1 + (-3.77 + 3.77i)T - 83iT^{2} \)
89 \( 1 + 9.52T + 89T^{2} \)
97 \( 1 + (-7.80 + 7.80i)T - 97iT^{2} \)
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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

−9.029415639857108956352948426198, −8.135359789719221750149150042998, −7.33552008793748965606389832370, −6.28745632033314374656731228603, −5.71070322148720976156979180043, −5.01709389003249741918225866467, −3.96864816748570444768730544398, −2.90503908206931132916193497290, −1.92597517446484017110141475132, −0.65155858091450498107763905769, 1.34331053898742827907517485177, 2.39649508747830777851719555460, 3.32954777167736861188148209195, 4.28459166628550281866554803373, 5.42028040630265296910585547481, 5.95481003453569943570581017987, 6.90775893526712805492652863155, 7.48074963001771146730225863729, 8.307041830000986647004186378752, 9.564201378566447587561190971551

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