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Properties

Label 2-2340-15.8-c1-0-10
Degree $2$
Conductor $2340$
Sign $0.979 - 0.202i$
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  + (−1.23 − 1.86i)5-s + (−0.157 + 0.157i)7-s + 2.06i·11-s + (−0.707 − 0.707i)13-s + (−0.379 − 0.379i)17-s + 5.74i·19-s + (1.48 − 1.48i)23-s + (−1.92 + 4.61i)25-s − 1.06·29-s + 8.85·31-s + (0.487 + 0.0979i)35-s + (−0.221 + 0.221i)37-s − 1.85i·41-s + (3.87 + 3.87i)43-s + (−0.646 − 0.646i)47-s + ⋯
L(s)  = 1  + (−0.554 − 0.832i)5-s + (−0.0594 + 0.0594i)7-s + 0.622i·11-s + (−0.196 − 0.196i)13-s + (−0.0919 − 0.0919i)17-s + 1.31i·19-s + (0.309 − 0.309i)23-s + (−0.385 + 0.922i)25-s − 0.198·29-s + 1.59·31-s + (0.0824 + 0.0165i)35-s + (−0.0364 + 0.0364i)37-s − 0.288i·41-s + (0.590 + 0.590i)43-s + (−0.0943 − 0.0943i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.979 - 0.202i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.979 - 0.202i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2340\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 13\)
Sign: $0.979 - 0.202i$
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: \((2,\ 2340,\ (\ :1/2),\ 0.979 - 0.202i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.431961775\)
\(L(\frac12)\) \(\approx\) \(1.431961775\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.23 + 1.86i)T \)
13 \( 1 + (0.707 + 0.707i)T \)
good7 \( 1 + (0.157 - 0.157i)T - 7iT^{2} \)
11 \( 1 - 2.06iT - 11T^{2} \)
17 \( 1 + (0.379 + 0.379i)T + 17iT^{2} \)
19 \( 1 - 5.74iT - 19T^{2} \)
23 \( 1 + (-1.48 + 1.48i)T - 23iT^{2} \)
29 \( 1 + 1.06T + 29T^{2} \)
31 \( 1 - 8.85T + 31T^{2} \)
37 \( 1 + (0.221 - 0.221i)T - 37iT^{2} \)
41 \( 1 + 1.85iT - 41T^{2} \)
43 \( 1 + (-3.87 - 3.87i)T + 43iT^{2} \)
47 \( 1 + (0.646 + 0.646i)T + 47iT^{2} \)
53 \( 1 + (-5.31 + 5.31i)T - 53iT^{2} \)
59 \( 1 - 0.345T + 59T^{2} \)
61 \( 1 - 0.645T + 61T^{2} \)
67 \( 1 + (-7.32 + 7.32i)T - 67iT^{2} \)
71 \( 1 + 2.07iT - 71T^{2} \)
73 \( 1 + (-2.76 - 2.76i)T + 73iT^{2} \)
79 \( 1 - 0.527iT - 79T^{2} \)
83 \( 1 + (-3.39 + 3.39i)T - 83iT^{2} \)
89 \( 1 - 14.3T + 89T^{2} \)
97 \( 1 + (9.21 - 9.21i)T - 97iT^{2} \)
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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

−8.984271898529820462884222387956, −8.142037395068798703691407399126, −7.69964316595719021535836646701, −6.72672497159788333306201941679, −5.82247991330678012499612635105, −4.93028016792282859580382135885, −4.28224860433470560089423396165, −3.35670307464350407840021721269, −2.11486640469294505266297531233, −0.893961030646840085729902234467, 0.66419507742820285713939636715, 2.36516243590262966465126401975, 3.13156113074183259085780500092, 4.04304197042825801922822235583, 4.92204008007608847544096279619, 5.97038606661157682584720382138, 6.76906393374987577517927850627, 7.30441157289088112321989419647, 8.207489505624511970814236490772, 8.866246260797458002893469491217

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