Properties

Label 2-3640-1.1-c1-0-34
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  − 2.47·3-s − 5-s − 7-s + 3.11·9-s + 3.58·11-s + 13-s + 2.47·15-s − 7.70·17-s − 1.47·19-s + 2.47·21-s − 2.87·23-s + 25-s − 0.284·27-s + 8.29·29-s + 6.47·31-s − 8.87·33-s + 35-s − 7.75·37-s − 2.47·39-s − 4.34·41-s + 0.945·43-s − 3.11·45-s + 8.15·47-s + 49-s + 19.0·51-s + 5.17·53-s − 3.58·55-s + ⋯
L(s)  = 1  − 1.42·3-s − 0.447·5-s − 0.377·7-s + 1.03·9-s + 1.08·11-s + 0.277·13-s + 0.638·15-s − 1.86·17-s − 0.337·19-s + 0.539·21-s − 0.598·23-s + 0.200·25-s − 0.0546·27-s + 1.53·29-s + 1.16·31-s − 1.54·33-s + 0.169·35-s − 1.27·37-s − 0.395·39-s − 0.678·41-s + 0.144·43-s − 0.464·45-s + 1.18·47-s + 0.142·49-s + 2.66·51-s + 0.710·53-s − 0.483·55-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 \( 1 + 2.47T + 3T^{2} \)
11 \( 1 - 3.58T + 11T^{2} \)
17 \( 1 + 7.70T + 17T^{2} \)
19 \( 1 + 1.47T + 19T^{2} \)
23 \( 1 + 2.87T + 23T^{2} \)
29 \( 1 - 8.29T + 29T^{2} \)
31 \( 1 - 6.47T + 31T^{2} \)
37 \( 1 + 7.75T + 37T^{2} \)
41 \( 1 + 4.34T + 41T^{2} \)
43 \( 1 - 0.945T + 43T^{2} \)
47 \( 1 - 8.15T + 47T^{2} \)
53 \( 1 - 5.17T + 53T^{2} \)
59 \( 1 + 1.45T + 59T^{2} \)
61 \( 1 + 9.58T + 61T^{2} \)
67 \( 1 - 6.11T + 67T^{2} \)
71 \( 1 - 12.1T + 71T^{2} \)
73 \( 1 - 13.3T + 73T^{2} \)
79 \( 1 - 9.06T + 79T^{2} \)
83 \( 1 + 14.4T + 83T^{2} \)
89 \( 1 + 1.60T + 89T^{2} \)
97 \( 1 + 9.24T + 97T^{2} \)
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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

−8.314053787790308873324009933418, −7.01631279828185222672799976139, −6.55122751257959955769606582167, −6.19012579089818219402307887557, −5.09976910111768756783504393044, −4.41678277950667169391252262666, −3.76539612772925655093572598668, −2.43105821699929996494480473551, −1.09116086009730539222697759362, 0, 1.09116086009730539222697759362, 2.43105821699929996494480473551, 3.76539612772925655093572598668, 4.41678277950667169391252262666, 5.09976910111768756783504393044, 6.19012579089818219402307887557, 6.55122751257959955769606582167, 7.01631279828185222672799976139, 8.314053787790308873324009933418

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