Properties

Label 2-3640-1.1-c1-0-56
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  + 0.402·3-s − 5-s + 7-s − 2.83·9-s + 1.78·11-s − 13-s − 0.402·15-s + 7.18·17-s − 5.62·19-s + 0.402·21-s + 1.08·23-s + 25-s − 2.34·27-s − 8.01·29-s − 7.90·31-s + 0.718·33-s − 35-s − 9.53·37-s − 0.402·39-s + 1.73·41-s + 10.9·43-s + 2.83·45-s + 0.542·47-s + 49-s + 2.89·51-s + 6.30·53-s − 1.78·55-s + ⋯
L(s)  = 1  + 0.232·3-s − 0.447·5-s + 0.377·7-s − 0.945·9-s + 0.538·11-s − 0.277·13-s − 0.103·15-s + 1.74·17-s − 1.29·19-s + 0.0878·21-s + 0.226·23-s + 0.200·25-s − 0.452·27-s − 1.48·29-s − 1.41·31-s + 0.125·33-s − 0.169·35-s − 1.56·37-s − 0.0644·39-s + 0.270·41-s + 1.67·43-s + 0.423·45-s + 0.0791·47-s + 0.142·49-s + 0.404·51-s + 0.866·53-s − 0.240·55-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 \( 1 - 0.402T + 3T^{2} \)
11 \( 1 - 1.78T + 11T^{2} \)
17 \( 1 - 7.18T + 17T^{2} \)
19 \( 1 + 5.62T + 19T^{2} \)
23 \( 1 - 1.08T + 23T^{2} \)
29 \( 1 + 8.01T + 29T^{2} \)
31 \( 1 + 7.90T + 31T^{2} \)
37 \( 1 + 9.53T + 37T^{2} \)
41 \( 1 - 1.73T + 41T^{2} \)
43 \( 1 - 10.9T + 43T^{2} \)
47 \( 1 - 0.542T + 47T^{2} \)
53 \( 1 - 6.30T + 53T^{2} \)
59 \( 1 + 1.12T + 59T^{2} \)
61 \( 1 + 2.71T + 61T^{2} \)
67 \( 1 + 10.8T + 67T^{2} \)
71 \( 1 + 4.26T + 71T^{2} \)
73 \( 1 - 14.1T + 73T^{2} \)
79 \( 1 + 0.819T + 79T^{2} \)
83 \( 1 + 0.910T + 83T^{2} \)
89 \( 1 + 9.09T + 89T^{2} \)
97 \( 1 + 10.2T + 97T^{2} \)
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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

−8.088959564185993404114365656417, −7.59057847533612610582993564964, −6.81258342349936927036531829118, −5.70053216603378096346573738263, −5.36274315641329399006502804941, −4.10292478336619107641313114002, −3.56280013296045319663965793437, −2.55634005837277883175771090461, −1.48786643616585563309624968970, 0, 1.48786643616585563309624968970, 2.55634005837277883175771090461, 3.56280013296045319663965793437, 4.10292478336619107641313114002, 5.36274315641329399006502804941, 5.70053216603378096346573738263, 6.81258342349936927036531829118, 7.59057847533612610582993564964, 8.088959564185993404114365656417

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