Properties

Label 2-3640-1.1-c1-0-65
Degree $2$
Conductor $3640$
Sign $-1$
Analytic cond. $29.0655$
Root an. cond. $5.39124$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.30·3-s + 5-s + 7-s − 1.30·9-s − 4.60·11-s − 13-s + 1.30·15-s + 6.30·17-s − 6.30·19-s + 1.30·21-s − 8·23-s + 25-s − 5.60·27-s + 1.90·29-s − 0.697·31-s − 6·33-s + 35-s − 3.90·37-s − 1.30·39-s − 4.30·41-s − 2·43-s − 1.30·45-s − 4.60·47-s + 49-s + 8.21·51-s − 2.60·53-s − 4.60·55-s + ⋯
L(s)  = 1  + 0.752·3-s + 0.447·5-s + 0.377·7-s − 0.434·9-s − 1.38·11-s − 0.277·13-s + 0.336·15-s + 1.52·17-s − 1.44·19-s + 0.284·21-s − 1.66·23-s + 0.200·25-s − 1.07·27-s + 0.354·29-s − 0.125·31-s − 1.04·33-s + 0.169·35-s − 0.642·37-s − 0.208·39-s − 0.671·41-s − 0.304·43-s − 0.194·45-s − 0.671·47-s + 0.142·49-s + 1.14·51-s − 0.357·53-s − 0.621·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3640\)    =    \(2^{3} \cdot 5 \cdot 7 \cdot 13\)
Sign: $-1$
Analytic conductor: \(29.0655\)
Root analytic conductor: \(5.39124\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3640} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 3640,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
7 \( 1 - T \)
13 \( 1 + T \)
good3 \( 1 - 1.30T + 3T^{2} \)
11 \( 1 + 4.60T + 11T^{2} \)
17 \( 1 - 6.30T + 17T^{2} \)
19 \( 1 + 6.30T + 19T^{2} \)
23 \( 1 + 8T + 23T^{2} \)
29 \( 1 - 1.90T + 29T^{2} \)
31 \( 1 + 0.697T + 31T^{2} \)
37 \( 1 + 3.90T + 37T^{2} \)
41 \( 1 + 4.30T + 41T^{2} \)
43 \( 1 + 2T + 43T^{2} \)
47 \( 1 + 4.60T + 47T^{2} \)
53 \( 1 + 2.60T + 53T^{2} \)
59 \( 1 - 11.1T + 59T^{2} \)
61 \( 1 + 1.21T + 61T^{2} \)
67 \( 1 - 8.90T + 67T^{2} \)
71 \( 1 + 2.60T + 71T^{2} \)
73 \( 1 + 8.60T + 73T^{2} \)
79 \( 1 - 3.51T + 79T^{2} \)
83 \( 1 + 11.8T + 83T^{2} \)
89 \( 1 + 7.30T + 89T^{2} \)
97 \( 1 + 13.2T + 97T^{2} \)
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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

−8.202291771525817068894475017954, −7.75544647959697324341432424762, −6.73912472484116101836529748356, −5.71966482278085944371779299034, −5.32770573608866530406381175695, −4.25637542501674371939580414143, −3.28428620990325479946114933008, −2.48758221826284606180735536989, −1.77580286047578491443949644123, 0, 1.77580286047578491443949644123, 2.48758221826284606180735536989, 3.28428620990325479946114933008, 4.25637542501674371939580414143, 5.32770573608866530406381175695, 5.71966482278085944371779299034, 6.73912472484116101836529748356, 7.75544647959697324341432424762, 8.202291771525817068894475017954

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