Properties

Label 2-3040-1.1-c1-0-60
Degree $2$
Conductor $3040$
Sign $-1$
Analytic cond. $24.2745$
Root an. cond. $4.92691$
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  + 0.539·3-s + 5-s + 0.630·7-s − 2.70·9-s − 1.70·11-s − 3.17·13-s + 0.539·15-s + 1.41·17-s + 19-s + 0.340·21-s − 4.04·23-s + 25-s − 3.07·27-s + 3.75·29-s + 1.41·31-s − 0.921·33-s + 0.630·35-s − 0.986·37-s − 1.70·39-s − 9.26·41-s + 5.70·43-s − 2.70·45-s − 4.04·47-s − 6.60·49-s + 0.764·51-s − 7.32·53-s − 1.70·55-s + ⋯
L(s)  = 1  + 0.311·3-s + 0.447·5-s + 0.238·7-s − 0.903·9-s − 0.515·11-s − 0.879·13-s + 0.139·15-s + 0.344·17-s + 0.229·19-s + 0.0742·21-s − 0.844·23-s + 0.200·25-s − 0.592·27-s + 0.697·29-s + 0.254·31-s − 0.160·33-s + 0.106·35-s − 0.162·37-s − 0.273·39-s − 1.44·41-s + 0.870·43-s − 0.403·45-s − 0.590·47-s − 0.943·49-s + 0.107·51-s − 1.00·53-s − 0.230·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3040\)    =    \(2^{5} \cdot 5 \cdot 19\)
Sign: $-1$
Analytic conductor: \(24.2745\)
Root analytic conductor: \(4.92691\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 3040,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
19 \( 1 - T \)
good3 \( 1 - 0.539T + 3T^{2} \)
7 \( 1 - 0.630T + 7T^{2} \)
11 \( 1 + 1.70T + 11T^{2} \)
13 \( 1 + 3.17T + 13T^{2} \)
17 \( 1 - 1.41T + 17T^{2} \)
23 \( 1 + 4.04T + 23T^{2} \)
29 \( 1 - 3.75T + 29T^{2} \)
31 \( 1 - 1.41T + 31T^{2} \)
37 \( 1 + 0.986T + 37T^{2} \)
41 \( 1 + 9.26T + 41T^{2} \)
43 \( 1 - 5.70T + 43T^{2} \)
47 \( 1 + 4.04T + 47T^{2} \)
53 \( 1 + 7.32T + 53T^{2} \)
59 \( 1 + 13.0T + 59T^{2} \)
61 \( 1 + 13.1T + 61T^{2} \)
67 \( 1 - 6.14T + 67T^{2} \)
71 \( 1 - 7.94T + 71T^{2} \)
73 \( 1 + 9.91T + 73T^{2} \)
79 \( 1 - 5.02T + 79T^{2} \)
83 \( 1 - 3.86T + 83T^{2} \)
89 \( 1 - 5.60T + 89T^{2} \)
97 \( 1 + 0.275T + 97T^{2} \)
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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

−8.092503522388348298958147397219, −7.937628267191987964791619905102, −6.80623410885709461557086029279, −6.01656440300720097313071234779, −5.24886201715245999558166971704, −4.58730037381520938106215440954, −3.30604687578726080251411384349, −2.64568692671073515884323644331, −1.66676547351451847819791366123, 0, 1.66676547351451847819791366123, 2.64568692671073515884323644331, 3.30604687578726080251411384349, 4.58730037381520938106215440954, 5.24886201715245999558166971704, 6.01656440300720097313071234779, 6.80623410885709461557086029279, 7.937628267191987964791619905102, 8.092503522388348298958147397219

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