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Properties

Label 2-2340-15.8-c1-0-18
Degree $2$
Conductor $2340$
Sign $0.574 + 0.818i$
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  + (2.17 + 0.520i)5-s + (−3.20 + 3.20i)7-s − 5.96i·11-s + (0.707 + 0.707i)13-s + (−1.36 − 1.36i)17-s − 5.65i·19-s + (1.98 − 1.98i)23-s + (4.45 + 2.26i)25-s − 0.153·29-s − 5.22·31-s + (−8.65 + 5.31i)35-s + (2.90 − 2.90i)37-s − 9.43i·41-s + (−5.30 − 5.30i)43-s + (6.90 + 6.90i)47-s + ⋯
L(s)  = 1  + (0.972 + 0.232i)5-s + (−1.21 + 1.21i)7-s − 1.79i·11-s + (0.196 + 0.196i)13-s + (−0.330 − 0.330i)17-s − 1.29i·19-s + (0.413 − 0.413i)23-s + (0.891 + 0.452i)25-s − 0.0285·29-s − 0.939·31-s + (−1.46 + 0.897i)35-s + (0.477 − 0.477i)37-s − 1.47i·41-s + (−0.809 − 0.809i)43-s + (1.00 + 1.00i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.574 + 0.818i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.574 + 0.818i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2340\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 13\)
Sign: $0.574 + 0.818i$
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: \((2,\ 2340,\ (\ :1/2),\ 0.574 + 0.818i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.570398428\)
\(L(\frac12)\) \(\approx\) \(1.570398428\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.17 - 0.520i)T \)
13 \( 1 + (-0.707 - 0.707i)T \)
good7 \( 1 + (3.20 - 3.20i)T - 7iT^{2} \)
11 \( 1 + 5.96iT - 11T^{2} \)
17 \( 1 + (1.36 + 1.36i)T + 17iT^{2} \)
19 \( 1 + 5.65iT - 19T^{2} \)
23 \( 1 + (-1.98 + 1.98i)T - 23iT^{2} \)
29 \( 1 + 0.153T + 29T^{2} \)
31 \( 1 + 5.22T + 31T^{2} \)
37 \( 1 + (-2.90 + 2.90i)T - 37iT^{2} \)
41 \( 1 + 9.43iT - 41T^{2} \)
43 \( 1 + (5.30 + 5.30i)T + 43iT^{2} \)
47 \( 1 + (-6.90 - 6.90i)T + 47iT^{2} \)
53 \( 1 + (-2.36 + 2.36i)T - 53iT^{2} \)
59 \( 1 - 5.88T + 59T^{2} \)
61 \( 1 - 13.5T + 61T^{2} \)
67 \( 1 + (0.187 - 0.187i)T - 67iT^{2} \)
71 \( 1 - 10.1iT - 71T^{2} \)
73 \( 1 + (-8.37 - 8.37i)T + 73iT^{2} \)
79 \( 1 + 14.0iT - 79T^{2} \)
83 \( 1 + (-5.88 + 5.88i)T - 83iT^{2} \)
89 \( 1 + 5.27T + 89T^{2} \)
97 \( 1 + (9.27 - 9.27i)T - 97iT^{2} \)
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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

−9.039305906619932770983007065919, −8.479223798605014222788193475102, −6.97940282641940182710733140622, −6.52869768514398538081016514463, −5.62871252214295641188915917574, −5.38392728778635812015729432004, −3.76311640011975465189302061843, −2.85834834333035427877226631343, −2.32117164932657933744406734046, −0.56259717699531578750944632742, 1.21130187873197711924276066834, 2.21139131485271074125444288657, 3.45325819177989774041457350697, 4.23449417199510617244431662510, 5.15160430832256168775039611985, 6.13248912389033560963669378331, 6.79000115054822910584143305389, 7.37244717500883174073720973990, 8.333627392127590357809083876140, 9.494218687096617881801953305147

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