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Properties

Label 2-2340-15.2-c1-0-2
Degree $2$
Conductor $2340$
Sign $0.590 - 0.806i$
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  + (−1.17 − 1.90i)5-s + (−0.865 − 0.865i)7-s + 3.06i·11-s + (0.707 − 0.707i)13-s + (−3.38 + 3.38i)17-s + 1.24i·19-s + (−3.64 − 3.64i)23-s + (−2.22 + 4.47i)25-s + 5.13·29-s + 8.10·31-s + (−0.624 + 2.66i)35-s + (−2.71 − 2.71i)37-s + 12.0i·41-s + (−2.22 + 2.22i)43-s + (−5.86 + 5.86i)47-s + ⋯
L(s)  = 1  + (−0.526 − 0.849i)5-s + (−0.327 − 0.327i)7-s + 0.924i·11-s + (0.196 − 0.196i)13-s + (−0.820 + 0.820i)17-s + 0.286i·19-s + (−0.759 − 0.759i)23-s + (−0.444 + 0.895i)25-s + 0.954·29-s + 1.45·31-s + (−0.105 + 0.450i)35-s + (−0.445 − 0.445i)37-s + 1.88i·41-s + (−0.339 + 0.339i)43-s + (−0.856 + 0.856i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.590 - 0.806i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.590 - 0.806i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2340\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 13\)
Sign: $0.590 - 0.806i$
Analytic conductor: \(18.6849\)
Root analytic conductor: \(4.32261\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2340} (1457, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2340,\ (\ :1/2),\ 0.590 - 0.806i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.082756260\)
\(L(\frac12)\) \(\approx\) \(1.082756260\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.17 + 1.90i)T \)
13 \( 1 + (-0.707 + 0.707i)T \)
good7 \( 1 + (0.865 + 0.865i)T + 7iT^{2} \)
11 \( 1 - 3.06iT - 11T^{2} \)
17 \( 1 + (3.38 - 3.38i)T - 17iT^{2} \)
19 \( 1 - 1.24iT - 19T^{2} \)
23 \( 1 + (3.64 + 3.64i)T + 23iT^{2} \)
29 \( 1 - 5.13T + 29T^{2} \)
31 \( 1 - 8.10T + 31T^{2} \)
37 \( 1 + (2.71 + 2.71i)T + 37iT^{2} \)
41 \( 1 - 12.0iT - 41T^{2} \)
43 \( 1 + (2.22 - 2.22i)T - 43iT^{2} \)
47 \( 1 + (5.86 - 5.86i)T - 47iT^{2} \)
53 \( 1 + (-7.51 - 7.51i)T + 53iT^{2} \)
59 \( 1 + 8.12T + 59T^{2} \)
61 \( 1 - 7.13T + 61T^{2} \)
67 \( 1 + (-7.19 - 7.19i)T + 67iT^{2} \)
71 \( 1 + 15.5iT - 71T^{2} \)
73 \( 1 + (-4.63 + 4.63i)T - 73iT^{2} \)
79 \( 1 - 11.2iT - 79T^{2} \)
83 \( 1 + (-0.0469 - 0.0469i)T + 83iT^{2} \)
89 \( 1 - 9.53T + 89T^{2} \)
97 \( 1 + (-9.95 - 9.95i)T + 97iT^{2} \)
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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

−9.014709717970705440458648092801, −8.242530142238645799349974304524, −7.80457169342183076052043770038, −6.67573863805818096857474576631, −6.15649671488666170874068859427, −4.83501735018810359989501924570, −4.43189628365997553416551223042, −3.52352822346902221438179949426, −2.24525033111579726210202265936, −1.03216897559143040617668373093, 0.43390513709862588628583208294, 2.22168746864976639644792318367, 3.09518491667311758609718899186, 3.83450567019170517823198685125, 4.87674791266978344401724460279, 5.89228819032203919969209128134, 6.63179891577207852011177839376, 7.18024889961741363032845377685, 8.253781539518218037266705625283, 8.687962591953698595413337607263

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