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Properties

Label 2-2340-15.8-c1-0-22
Degree $2$
Conductor $2340$
Sign $0.0637 + 0.997i$
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  + (2.01 − 0.967i)5-s + (3.08 − 3.08i)7-s + 0.249i·11-s + (−0.707 − 0.707i)13-s + (−5.12 − 5.12i)17-s + 1.73i·19-s + (2.74 − 2.74i)23-s + (3.12 − 3.90i)25-s + 1.33·29-s + 3.60·31-s + (3.23 − 9.20i)35-s + (0.612 − 0.612i)37-s + 4.09i·41-s + (−7.55 − 7.55i)43-s + (−4.43 − 4.43i)47-s + ⋯
L(s)  = 1  + (0.901 − 0.432i)5-s + (1.16 − 1.16i)7-s + 0.0753i·11-s + (−0.196 − 0.196i)13-s + (−1.24 − 1.24i)17-s + 0.397i·19-s + (0.573 − 0.573i)23-s + (0.625 − 0.780i)25-s + 0.248·29-s + 0.646·31-s + (0.546 − 1.55i)35-s + (0.100 − 0.100i)37-s + 0.639i·41-s + (−1.15 − 1.15i)43-s + (−0.647 − 0.647i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0637 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0637 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2340\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 13\)
Sign: $0.0637 + 0.997i$
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: \((2,\ 2340,\ (\ :1/2),\ 0.0637 + 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.248358315\)
\(L(\frac12)\) \(\approx\) \(2.248358315\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.01 + 0.967i)T \)
13 \( 1 + (0.707 + 0.707i)T \)
good7 \( 1 + (-3.08 + 3.08i)T - 7iT^{2} \)
11 \( 1 - 0.249iT - 11T^{2} \)
17 \( 1 + (5.12 + 5.12i)T + 17iT^{2} \)
19 \( 1 - 1.73iT - 19T^{2} \)
23 \( 1 + (-2.74 + 2.74i)T - 23iT^{2} \)
29 \( 1 - 1.33T + 29T^{2} \)
31 \( 1 - 3.60T + 31T^{2} \)
37 \( 1 + (-0.612 + 0.612i)T - 37iT^{2} \)
41 \( 1 - 4.09iT - 41T^{2} \)
43 \( 1 + (7.55 + 7.55i)T + 43iT^{2} \)
47 \( 1 + (4.43 + 4.43i)T + 47iT^{2} \)
53 \( 1 + (9.54 - 9.54i)T - 53iT^{2} \)
59 \( 1 - 1.05T + 59T^{2} \)
61 \( 1 - 8.93T + 61T^{2} \)
67 \( 1 + (3.67 - 3.67i)T - 67iT^{2} \)
71 \( 1 - 9.24iT - 71T^{2} \)
73 \( 1 + (-10.5 - 10.5i)T + 73iT^{2} \)
79 \( 1 - 4.35iT - 79T^{2} \)
83 \( 1 + (6.84 - 6.84i)T - 83iT^{2} \)
89 \( 1 + 7.38T + 89T^{2} \)
97 \( 1 + (-11.4 + 11.4i)T - 97iT^{2} \)
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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

−8.668927283645947550256220250845, −8.196514543725362541820866535874, −7.12208836262764099788354104302, −6.68413611440947192208955209976, −5.46006020907054149578524868507, −4.77985432602798467986859306088, −4.25014661375694878725053007947, −2.79807311585403445682960495654, −1.78572556496987381266641136086, −0.75721424499316034759943632258, 1.64380558701526234830226102576, 2.20918812413342646717876167859, 3.24460537486031950379321549929, 4.67270942245983180457955370086, 5.14814516598845891726463927026, 6.17349397839642864575605304835, 6.58810009629466737586724567982, 7.77883313179698516489459202446, 8.513124809328105050125763778136, 9.084805848250467359032436713985

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