Properties

Label 2-3040-1.1-c1-0-8
Degree $2$
Conductor $3040$
Sign $1$
Analytic cond. $24.2745$
Root an. cond. $4.92691$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.08·3-s − 5-s + 0.648·7-s + 6.52·9-s − 1.35·11-s + 4.43·13-s + 3.08·15-s + 2·17-s + 19-s − 2·21-s + 3.35·23-s + 25-s − 10.8·27-s − 4.17·29-s − 4.17·31-s + 4.17·33-s − 0.648·35-s − 2.43·37-s − 13.6·39-s + 10.1·41-s − 6.82·43-s − 6.52·45-s + 0.648·47-s − 6.58·49-s − 6.17·51-s + 6.43·53-s + 1.35·55-s + ⋯
L(s)  = 1  − 1.78·3-s − 0.447·5-s + 0.244·7-s + 2.17·9-s − 0.407·11-s + 1.23·13-s + 0.796·15-s + 0.485·17-s + 0.229·19-s − 0.436·21-s + 0.698·23-s + 0.200·25-s − 2.09·27-s − 0.774·29-s − 0.749·31-s + 0.726·33-s − 0.109·35-s − 0.400·37-s − 2.19·39-s + 1.58·41-s − 1.04·43-s − 0.972·45-s + 0.0945·47-s − 0.940·49-s − 0.864·51-s + 0.884·53-s + 0.182·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3040\)    =    \(2^{5} \cdot 5 \cdot 19\)
Sign: $1$
Analytic conductor: \(24.2745\)
Root analytic conductor: \(4.92691\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3040,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8523397991\)
\(L(\frac12)\) \(\approx\) \(0.8523397991\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + T \)
19 \( 1 - T \)
good3 \( 1 + 3.08T + 3T^{2} \)
7 \( 1 - 0.648T + 7T^{2} \)
11 \( 1 + 1.35T + 11T^{2} \)
13 \( 1 - 4.43T + 13T^{2} \)
17 \( 1 - 2T + 17T^{2} \)
23 \( 1 - 3.35T + 23T^{2} \)
29 \( 1 + 4.17T + 29T^{2} \)
31 \( 1 + 4.17T + 31T^{2} \)
37 \( 1 + 2.43T + 37T^{2} \)
41 \( 1 - 10.1T + 41T^{2} \)
43 \( 1 + 6.82T + 43T^{2} \)
47 \( 1 - 0.648T + 47T^{2} \)
53 \( 1 - 6.43T + 53T^{2} \)
59 \( 1 + 10.3T + 59T^{2} \)
61 \( 1 + 2.11T + 61T^{2} \)
67 \( 1 - 12.1T + 67T^{2} \)
71 \( 1 - 3.82T + 71T^{2} \)
73 \( 1 + 4.17T + 73T^{2} \)
79 \( 1 + 2T + 79T^{2} \)
83 \( 1 + 9.52T + 83T^{2} \)
89 \( 1 - 3.82T + 89T^{2} \)
97 \( 1 - 7.73T + 97T^{2} \)
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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

−8.692814015348887199836474227932, −7.74576552227558984792912644017, −7.12676371782241235200987184536, −6.29693301520996344656435156335, −5.64415951115799450045803361617, −5.05143929559786029579676372545, −4.20062744463933678767525474755, −3.32428631479572379754329648161, −1.63804827902822657248179707060, −0.64709512870536653825115727659, 0.64709512870536653825115727659, 1.63804827902822657248179707060, 3.32428631479572379754329648161, 4.20062744463933678767525474755, 5.05143929559786029579676372545, 5.64415951115799450045803361617, 6.29693301520996344656435156335, 7.12676371782241235200987184536, 7.74576552227558984792912644017, 8.692814015348887199836474227932

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