Properties

Label 2-3040-1.1-c1-0-63
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  + 1.67·3-s + 5-s − 4.15·7-s − 0.193·9-s + 0.806·11-s + 0.481·13-s + 1.67·15-s − 3.61·17-s + 19-s − 6.96·21-s + 5.76·23-s + 25-s − 5.35·27-s − 8.57·29-s − 3.61·31-s + 1.35·33-s − 4.15·35-s − 9.18·37-s + 0.806·39-s + 0.312·41-s + 3.19·43-s − 0.193·45-s + 5.76·47-s + 10.2·49-s − 6.05·51-s − 8.21·53-s + 0.806·55-s + ⋯
L(s)  = 1  + 0.967·3-s + 0.447·5-s − 1.57·7-s − 0.0646·9-s + 0.243·11-s + 0.133·13-s + 0.432·15-s − 0.876·17-s + 0.229·19-s − 1.51·21-s + 1.20·23-s + 0.200·25-s − 1.02·27-s − 1.59·29-s − 0.648·31-s + 0.235·33-s − 0.702·35-s − 1.50·37-s + 0.129·39-s + 0.0488·41-s + 0.487·43-s − 0.0289·45-s + 0.841·47-s + 1.46·49-s − 0.847·51-s − 1.12·53-s + 0.108·55-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 \( 1 - 1.67T + 3T^{2} \)
7 \( 1 + 4.15T + 7T^{2} \)
11 \( 1 - 0.806T + 11T^{2} \)
13 \( 1 - 0.481T + 13T^{2} \)
17 \( 1 + 3.61T + 17T^{2} \)
23 \( 1 - 5.76T + 23T^{2} \)
29 \( 1 + 8.57T + 29T^{2} \)
31 \( 1 + 3.61T + 31T^{2} \)
37 \( 1 + 9.18T + 37T^{2} \)
41 \( 1 - 0.312T + 41T^{2} \)
43 \( 1 - 3.19T + 43T^{2} \)
47 \( 1 - 5.76T + 47T^{2} \)
53 \( 1 + 8.21T + 53T^{2} \)
59 \( 1 - 8.88T + 59T^{2} \)
61 \( 1 + 5.58T + 61T^{2} \)
67 \( 1 + 9.59T + 67T^{2} \)
71 \( 1 + 16.2T + 71T^{2} \)
73 \( 1 + 2.12T + 73T^{2} \)
79 \( 1 + 16.8T + 79T^{2} \)
83 \( 1 - 5.89T + 83T^{2} \)
89 \( 1 + 11.2T + 89T^{2} \)
97 \( 1 - 17.4T + 97T^{2} \)
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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

−8.713783753906031747633844825076, −7.45134239399546305259248176562, −6.97041803190922164229463500413, −6.08518322118070209548618230151, −5.43933394021951906555524099045, −4.14685202017847989071547698720, −3.32300772674790813735121862622, −2.79115382641516411144824972807, −1.74188107202953454702611274082, 0, 1.74188107202953454702611274082, 2.79115382641516411144824972807, 3.32300772674790813735121862622, 4.14685202017847989071547698720, 5.43933394021951906555524099045, 6.08518322118070209548618230151, 6.97041803190922164229463500413, 7.45134239399546305259248176562, 8.713783753906031747633844825076

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