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Properties

Label 2-2340-15.8-c1-0-19
Degree $2$
Conductor $2340$
Sign $0.831 + 0.555i$
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  + (2.21 − 0.300i)5-s + (2.83 − 2.83i)7-s − 0.275i·11-s + (0.707 + 0.707i)13-s + (1.49 + 1.49i)17-s − 0.767i·19-s + (−2.48 + 2.48i)23-s + (4.81 − 1.33i)25-s + 9.19·29-s + 0.0656·31-s + (5.43 − 7.13i)35-s + (2.85 − 2.85i)37-s + 3.86i·41-s + (1.63 + 1.63i)43-s + (−2.80 − 2.80i)47-s + ⋯
L(s)  = 1  + (0.990 − 0.134i)5-s + (1.07 − 1.07i)7-s − 0.0829i·11-s + (0.196 + 0.196i)13-s + (0.361 + 0.361i)17-s − 0.176i·19-s + (−0.518 + 0.518i)23-s + (0.963 − 0.266i)25-s + 1.70·29-s + 0.0117·31-s + (0.918 − 1.20i)35-s + (0.468 − 0.468i)37-s + 0.603i·41-s + (0.250 + 0.250i)43-s + (−0.408 − 0.408i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.831 + 0.555i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.831 + 0.555i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2340\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 13\)
Sign: $0.831 + 0.555i$
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: \((2,\ 2340,\ (\ :1/2),\ 0.831 + 0.555i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.602895440\)
\(L(\frac12)\) \(\approx\) \(2.602895440\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.21 + 0.300i)T \)
13 \( 1 + (-0.707 - 0.707i)T \)
good7 \( 1 + (-2.83 + 2.83i)T - 7iT^{2} \)
11 \( 1 + 0.275iT - 11T^{2} \)
17 \( 1 + (-1.49 - 1.49i)T + 17iT^{2} \)
19 \( 1 + 0.767iT - 19T^{2} \)
23 \( 1 + (2.48 - 2.48i)T - 23iT^{2} \)
29 \( 1 - 9.19T + 29T^{2} \)
31 \( 1 - 0.0656T + 31T^{2} \)
37 \( 1 + (-2.85 + 2.85i)T - 37iT^{2} \)
41 \( 1 - 3.86iT - 41T^{2} \)
43 \( 1 + (-1.63 - 1.63i)T + 43iT^{2} \)
47 \( 1 + (2.80 + 2.80i)T + 47iT^{2} \)
53 \( 1 + (-0.976 + 0.976i)T - 53iT^{2} \)
59 \( 1 + 7.13T + 59T^{2} \)
61 \( 1 + 9.52T + 61T^{2} \)
67 \( 1 + (-0.223 + 0.223i)T - 67iT^{2} \)
71 \( 1 - 12.7iT - 71T^{2} \)
73 \( 1 + (9.60 + 9.60i)T + 73iT^{2} \)
79 \( 1 + 5.20iT - 79T^{2} \)
83 \( 1 + (-5.65 + 5.65i)T - 83iT^{2} \)
89 \( 1 - 5.96T + 89T^{2} \)
97 \( 1 + (-0.0852 + 0.0852i)T - 97iT^{2} \)
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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

−8.882763494457194582026548930545, −8.128000404377504903835902305867, −7.45867249704636441142129196831, −6.53009459967038841578726351252, −5.81970162518598453770552062678, −4.84377093777173840616425846240, −4.27777228415714507190450635156, −3.07022381595051367157201683753, −1.83567249187795467906568027226, −1.04774286536668255464080409382, 1.28930075115010406169930561127, 2.26061639718174023167882648349, 3.00235482302845880564219219642, 4.49617776577390919865877503311, 5.16291481757617414140666882106, 5.91233925110585936639707235906, 6.53251988025173496849733257079, 7.67051267614315291815661160688, 8.389015933995713203701243875552, 8.993820881822656122700209948353

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