Properties

Label 2-3610-1.1-c1-0-10
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 − 3.21·3-s + 4-s + 5-s + 3.21·6-s − 0.0233·7-s − 8-s + 7.34·9-s − 10-s + 2.16·11-s − 3.21·12-s − 1.59·13-s + 0.0233·14-s − 3.21·15-s + 16-s − 7.97·17-s − 7.34·18-s + 20-s + 0.0752·21-s − 2.16·22-s + 4.48·23-s + 3.21·24-s + 25-s + 1.59·26-s − 13.9·27-s − 0.0233·28-s + 7.55·29-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.85·3-s + 0.5·4-s + 0.447·5-s + 1.31·6-s − 0.00884·7-s − 0.353·8-s + 2.44·9-s − 0.316·10-s + 0.651·11-s − 0.928·12-s − 0.441·13-s + 0.00625·14-s − 0.830·15-s + 0.250·16-s − 1.93·17-s − 1.73·18-s + 0.223·20-s + 0.0164·21-s − 0.460·22-s + 0.934·23-s + 0.656·24-s + 0.200·25-s + 0.311·26-s − 2.68·27-s − 0.00442·28-s + 1.40·29-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

\(L(1)\) \(\approx\) \(0.6497228328\)
\(L(\frac12)\) \(\approx\) \(0.6497228328\)
\(L(\frac{3}{2})\) 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 \( 1 + 3.21T + 3T^{2} \)
7 \( 1 + 0.0233T + 7T^{2} \)
11 \( 1 - 2.16T + 11T^{2} \)
13 \( 1 + 1.59T + 13T^{2} \)
17 \( 1 + 7.97T + 17T^{2} \)
23 \( 1 - 4.48T + 23T^{2} \)
29 \( 1 - 7.55T + 29T^{2} \)
31 \( 1 - 4.45T + 31T^{2} \)
37 \( 1 + 0.389T + 37T^{2} \)
41 \( 1 - 5.60T + 41T^{2} \)
43 \( 1 - 9.70T + 43T^{2} \)
47 \( 1 - 4.27T + 47T^{2} \)
53 \( 1 + 3.78T + 53T^{2} \)
59 \( 1 + 3.63T + 59T^{2} \)
61 \( 1 + 11.5T + 61T^{2} \)
67 \( 1 - 3.85T + 67T^{2} \)
71 \( 1 + 12.8T + 71T^{2} \)
73 \( 1 + 1.90T + 73T^{2} \)
79 \( 1 + 9.51T + 79T^{2} \)
83 \( 1 + 2.62T + 83T^{2} \)
89 \( 1 - 8.87T + 89T^{2} \)
97 \( 1 - 11.1T + 97T^{2} \)
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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

−8.785075672086052597988941373334, −7.56607775542748753108879899436, −6.83743204378689136184106057695, −6.38707427094561250274252081997, −5.82444187120280329955987095101, −4.71694576562605715915701254410, −4.39730291572183119394044561317, −2.71475510259387992665019541360, −1.55864457226251349293745109696, −0.60587022598474903503873201839, 0.60587022598474903503873201839, 1.55864457226251349293745109696, 2.71475510259387992665019541360, 4.39730291572183119394044561317, 4.71694576562605715915701254410, 5.82444187120280329955987095101, 6.38707427094561250274252081997, 6.83743204378689136184106057695, 7.56607775542748753108879899436, 8.785075672086052597988941373334

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