Properties

Label 2-3640-1.1-c1-0-61
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  + 1.93·3-s − 5-s − 7-s + 0.745·9-s − 3.18·11-s + 13-s − 1.93·15-s + 1.44·17-s + 2.93·19-s − 1.93·21-s − 0.173·23-s + 25-s − 4.36·27-s − 7.63·29-s + 2.06·31-s − 6.17·33-s + 35-s − 7.42·37-s + 1.93·39-s + 2.76·41-s − 7.87·43-s − 0.745·45-s + 9.53·47-s + 49-s + 2.79·51-s − 8.37·53-s + 3.18·55-s + ⋯
L(s)  = 1  + 1.11·3-s − 0.447·5-s − 0.377·7-s + 0.248·9-s − 0.961·11-s + 0.277·13-s − 0.499·15-s + 0.350·17-s + 0.673·19-s − 0.422·21-s − 0.0360·23-s + 0.200·25-s − 0.839·27-s − 1.41·29-s + 0.370·31-s − 1.07·33-s + 0.169·35-s − 1.22·37-s + 0.309·39-s + 0.431·41-s − 1.20·43-s − 0.111·45-s + 1.39·47-s + 0.142·49-s + 0.391·51-s − 1.15·53-s + 0.430·55-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: Trivial
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 \( 1 - 1.93T + 3T^{2} \)
11 \( 1 + 3.18T + 11T^{2} \)
17 \( 1 - 1.44T + 17T^{2} \)
19 \( 1 - 2.93T + 19T^{2} \)
23 \( 1 + 0.173T + 23T^{2} \)
29 \( 1 + 7.63T + 29T^{2} \)
31 \( 1 - 2.06T + 31T^{2} \)
37 \( 1 + 7.42T + 37T^{2} \)
41 \( 1 - 2.76T + 41T^{2} \)
43 \( 1 + 7.87T + 43T^{2} \)
47 \( 1 - 9.53T + 47T^{2} \)
53 \( 1 + 8.37T + 53T^{2} \)
59 \( 1 + 11.9T + 59T^{2} \)
61 \( 1 + 2.81T + 61T^{2} \)
67 \( 1 - 3.74T + 67T^{2} \)
71 \( 1 + 10.2T + 71T^{2} \)
73 \( 1 - 1.15T + 73T^{2} \)
79 \( 1 + 2.12T + 79T^{2} \)
83 \( 1 + 4.98T + 83T^{2} \)
89 \( 1 - 0.108T + 89T^{2} \)
97 \( 1 - 10.4T + 97T^{2} \)
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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

−8.021349354466805918150486519245, −7.72150700228805189067603700731, −6.93266209895763377376434603704, −5.85274355126107140094660419803, −5.16104141922768255329856625783, −4.05784279896069769615385405030, −3.29599462019995738371732590685, −2.76364799868910743741680339458, −1.64601193469162011330487859287, 0, 1.64601193469162011330487859287, 2.76364799868910743741680339458, 3.29599462019995738371732590685, 4.05784279896069769615385405030, 5.16104141922768255329856625783, 5.85274355126107140094660419803, 6.93266209895763377376434603704, 7.72150700228805189067603700731, 8.021349354466805918150486519245

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