Properties

Label 2-3610-1.1-c1-0-28
Degree $2$
Conductor $3610$
Sign $1$
Analytic cond. $28.8259$
Root an. cond. $5.36898$
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  − 2-s + 0.0361·3-s + 4-s + 5-s − 0.0361·6-s + 1.83·7-s − 8-s − 2.99·9-s − 10-s − 2.46·11-s + 0.0361·12-s + 2.39·13-s − 1.83·14-s + 0.0361·15-s + 16-s + 6.74·17-s + 2.99·18-s + 20-s + 0.0664·21-s + 2.46·22-s + 1.62·23-s − 0.0361·24-s + 25-s − 2.39·26-s − 0.216·27-s + 1.83·28-s + 3.30·29-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.0208·3-s + 0.5·4-s + 0.447·5-s − 0.0147·6-s + 0.694·7-s − 0.353·8-s − 0.999·9-s − 0.316·10-s − 0.743·11-s + 0.0104·12-s + 0.663·13-s − 0.491·14-s + 0.00933·15-s + 0.250·16-s + 1.63·17-s + 0.706·18-s + 0.223·20-s + 0.0144·21-s + 0.525·22-s + 0.337·23-s − 0.00737·24-s + 0.200·25-s − 0.469·26-s − 0.0417·27-s + 0.347·28-s + 0.613·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3610 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3610 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3610\)    =    \(2 \cdot 5 \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(28.8259\)
Root analytic conductor: \(5.36898\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3610,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.509627595\)
\(L(\frac12)\) \(\approx\) \(1.509627595\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
5 \( 1 - T \)
19 \( 1 \)
good3 \( 1 - 0.0361T + 3T^{2} \)
7 \( 1 - 1.83T + 7T^{2} \)
11 \( 1 + 2.46T + 11T^{2} \)
13 \( 1 - 2.39T + 13T^{2} \)
17 \( 1 - 6.74T + 17T^{2} \)
23 \( 1 - 1.62T + 23T^{2} \)
29 \( 1 - 3.30T + 29T^{2} \)
31 \( 1 + 3.50T + 31T^{2} \)
37 \( 1 + 6.00T + 37T^{2} \)
41 \( 1 + 7.98T + 41T^{2} \)
43 \( 1 - 5.71T + 43T^{2} \)
47 \( 1 - 12.4T + 47T^{2} \)
53 \( 1 + 1.92T + 53T^{2} \)
59 \( 1 + 4.75T + 59T^{2} \)
61 \( 1 - 13.6T + 61T^{2} \)
67 \( 1 + 3.71T + 67T^{2} \)
71 \( 1 - 5.83T + 71T^{2} \)
73 \( 1 + 0.510T + 73T^{2} \)
79 \( 1 - 12.1T + 79T^{2} \)
83 \( 1 - 6.23T + 83T^{2} \)
89 \( 1 - 17.0T + 89T^{2} \)
97 \( 1 + 13.6T + 97T^{2} \)
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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

−8.474242620164301160125352759085, −8.006671039326122961179303294023, −7.27957022252101705319296414679, −6.30598873530587942455225348543, −5.50547664813474427803528201739, −5.11759905996759599550044746690, −3.64747910827765926271864631173, −2.85313337124490118650283456613, −1.88687895112688601019880438531, −0.814305962051955273069140660837, 0.814305962051955273069140660837, 1.88687895112688601019880438531, 2.85313337124490118650283456613, 3.64747910827765926271864631173, 5.11759905996759599550044746690, 5.50547664813474427803528201739, 6.30598873530587942455225348543, 7.27957022252101705319296414679, 8.006671039326122961179303294023, 8.474242620164301160125352759085

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