Properties

Label 2-3640-1.1-c1-0-67
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  + 1.75·3-s + 5-s − 7-s + 0.0856·9-s − 3.23·11-s + 13-s + 1.75·15-s − 4.59·17-s + 0.150·19-s − 1.75·21-s + 1.40·23-s + 25-s − 5.11·27-s − 8.67·29-s − 4.96·31-s − 5.68·33-s − 35-s − 0.520·37-s + 1.75·39-s + 3.55·41-s + 0.277·43-s + 0.0856·45-s − 11.1·47-s + 49-s − 8.07·51-s + 0.171·53-s − 3.23·55-s + ⋯
L(s)  = 1  + 1.01·3-s + 0.447·5-s − 0.377·7-s + 0.0285·9-s − 0.975·11-s + 0.277·13-s + 0.453·15-s − 1.11·17-s + 0.0345·19-s − 0.383·21-s + 0.293·23-s + 0.200·25-s − 0.985·27-s − 1.61·29-s − 0.892·31-s − 0.989·33-s − 0.169·35-s − 0.0855·37-s + 0.281·39-s + 0.555·41-s + 0.0422·43-s + 0.0127·45-s − 1.62·47-s + 0.142·49-s − 1.13·51-s + 0.0235·53-s − 0.436·55-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 \( 1 - 1.75T + 3T^{2} \)
11 \( 1 + 3.23T + 11T^{2} \)
17 \( 1 + 4.59T + 17T^{2} \)
19 \( 1 - 0.150T + 19T^{2} \)
23 \( 1 - 1.40T + 23T^{2} \)
29 \( 1 + 8.67T + 29T^{2} \)
31 \( 1 + 4.96T + 31T^{2} \)
37 \( 1 + 0.520T + 37T^{2} \)
41 \( 1 - 3.55T + 41T^{2} \)
43 \( 1 - 0.277T + 43T^{2} \)
47 \( 1 + 11.1T + 47T^{2} \)
53 \( 1 - 0.171T + 53T^{2} \)
59 \( 1 - 7.78T + 59T^{2} \)
61 \( 1 - 3.68T + 61T^{2} \)
67 \( 1 + 1.95T + 67T^{2} \)
71 \( 1 + 1.55T + 71T^{2} \)
73 \( 1 - 5.85T + 73T^{2} \)
79 \( 1 + 5.00T + 79T^{2} \)
83 \( 1 + 10.3T + 83T^{2} \)
89 \( 1 + 15.4T + 89T^{2} \)
97 \( 1 - 0.869T + 97T^{2} \)
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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

−8.301753706830908611609618969551, −7.51541672750358880541433162137, −6.82970035480518123470351836421, −5.85383996586136146947694339092, −5.24586828626391151538739432626, −4.13560308621852390306455175429, −3.30382241927306121630789328208, −2.53277124020744035810263799466, −1.80488282532410733670693829321, 0, 1.80488282532410733670693829321, 2.53277124020744035810263799466, 3.30382241927306121630789328208, 4.13560308621852390306455175429, 5.24586828626391151538739432626, 5.85383996586136146947694339092, 6.82970035480518123470351836421, 7.51541672750358880541433162137, 8.301753706830908611609618969551

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