Properties

Label 2-3640-1.1-c1-0-40
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3.11·3-s − 5-s + 7-s + 6.72·9-s − 13-s − 3.11·15-s + 3.72·17-s + 1.72·19-s + 3.11·21-s + 1.21·23-s + 25-s + 11.6·27-s − 6.33·29-s + 7.11·31-s − 35-s + 3.90·37-s − 3.11·39-s + 3.72·41-s − 10.2·43-s − 6.72·45-s + 2.78·47-s + 49-s + 11.6·51-s + 2·53-s + 5.39·57-s + 13.3·59-s − 10·61-s + ⋯
L(s)  = 1  + 1.80·3-s − 0.447·5-s + 0.377·7-s + 2.24·9-s − 0.277·13-s − 0.805·15-s + 0.904·17-s + 0.396·19-s + 0.680·21-s + 0.254·23-s + 0.200·25-s + 2.23·27-s − 1.17·29-s + 1.27·31-s − 0.169·35-s + 0.641·37-s − 0.499·39-s + 0.582·41-s − 1.56·43-s − 1.00·45-s + 0.405·47-s + 0.142·49-s + 1.62·51-s + 0.274·53-s + 0.714·57-s + 1.73·59-s − 1.28·61-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: \(0\)
Selberg data: \((2,\ 3640,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.879591493\)
\(L(\frac12)\) \(\approx\) \(3.879591493\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.11T + 3T^{2} \)
11 \( 1 + 11T^{2} \)
17 \( 1 - 3.72T + 17T^{2} \)
19 \( 1 - 1.72T + 19T^{2} \)
23 \( 1 - 1.21T + 23T^{2} \)
29 \( 1 + 6.33T + 29T^{2} \)
31 \( 1 - 7.11T + 31T^{2} \)
37 \( 1 - 3.90T + 37T^{2} \)
41 \( 1 - 3.72T + 41T^{2} \)
43 \( 1 + 10.2T + 43T^{2} \)
47 \( 1 - 2.78T + 47T^{2} \)
53 \( 1 - 2T + 53T^{2} \)
59 \( 1 - 13.3T + 59T^{2} \)
61 \( 1 + 10T + 61T^{2} \)
67 \( 1 + 3.29T + 67T^{2} \)
71 \( 1 + 6.23T + 71T^{2} \)
73 \( 1 - 15.6T + 73T^{2} \)
79 \( 1 + 1.72T + 79T^{2} \)
83 \( 1 + 1.21T + 83T^{2} \)
89 \( 1 - 10.1T + 89T^{2} \)
97 \( 1 + 5.80T + 97T^{2} \)
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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

−8.472217050929470834781616696482, −7.80162321910971348930938607871, −7.49817768243142560319627084004, −6.59303621552810476118701737877, −5.36250824826443544376028398660, −4.47353853358568203516691275983, −3.70220862204438419341270941492, −3.03416869963661629802707739462, −2.20576192420505254849829142256, −1.14850584815261554197436153461, 1.14850584815261554197436153461, 2.20576192420505254849829142256, 3.03416869963661629802707739462, 3.70220862204438419341270941492, 4.47353853358568203516691275983, 5.36250824826443544376028398660, 6.59303621552810476118701737877, 7.49817768243142560319627084004, 7.80162321910971348930938607871, 8.472217050929470834781616696482

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