Properties

Label 2-7600-1.1-c1-0-65
Degree $2$
Conductor $7600$
Sign $1$
Analytic cond. $60.6863$
Root an. cond. $7.79014$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.23·3-s + 4.47·7-s − 1.47·9-s + 3.23·13-s + 6.47·17-s − 19-s − 5.52·21-s + 2·23-s + 5.52·27-s + 2·29-s + 1.52·31-s − 4.76·37-s − 4.00·39-s + 3.52·41-s − 0.472·43-s + 12.4·47-s + 13.0·49-s − 8.00·51-s − 11.2·53-s + 1.23·57-s + 10.4·59-s − 4.47·61-s − 6.58·63-s + 1.23·67-s − 2.47·69-s + 1.52·71-s − 6.47·73-s + ⋯
L(s)  = 1  − 0.713·3-s + 1.69·7-s − 0.490·9-s + 0.897·13-s + 1.56·17-s − 0.229·19-s − 1.20·21-s + 0.417·23-s + 1.06·27-s + 0.371·29-s + 0.274·31-s − 0.783·37-s − 0.640·39-s + 0.550·41-s − 0.0720·43-s + 1.81·47-s + 1.85·49-s − 1.12·51-s − 1.54·53-s + 0.163·57-s + 1.36·59-s − 0.572·61-s − 0.829·63-s + 0.151·67-s − 0.297·69-s + 0.181·71-s − 0.757·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7600\)    =    \(2^{4} \cdot 5^{2} \cdot 19\)
Sign: $1$
Analytic conductor: \(60.6863\)
Root analytic conductor: \(7.79014\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 7600,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.187588980\)
\(L(\frac12)\) \(\approx\) \(2.187588980\)
\(L(\frac{3}{2})\) 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 \( 1 \)
19 \( 1 + T \)
good3 \( 1 + 1.23T + 3T^{2} \)
7 \( 1 - 4.47T + 7T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 - 3.23T + 13T^{2} \)
17 \( 1 - 6.47T + 17T^{2} \)
23 \( 1 - 2T + 23T^{2} \)
29 \( 1 - 2T + 29T^{2} \)
31 \( 1 - 1.52T + 31T^{2} \)
37 \( 1 + 4.76T + 37T^{2} \)
41 \( 1 - 3.52T + 41T^{2} \)
43 \( 1 + 0.472T + 43T^{2} \)
47 \( 1 - 12.4T + 47T^{2} \)
53 \( 1 + 11.2T + 53T^{2} \)
59 \( 1 - 10.4T + 59T^{2} \)
61 \( 1 + 4.47T + 61T^{2} \)
67 \( 1 - 1.23T + 67T^{2} \)
71 \( 1 - 1.52T + 71T^{2} \)
73 \( 1 + 6.47T + 73T^{2} \)
79 \( 1 - 6.47T + 79T^{2} \)
83 \( 1 + 14.9T + 83T^{2} \)
89 \( 1 + 6.94T + 89T^{2} \)
97 \( 1 + 4.18T + 97T^{2} \)
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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

−7.87506370992720182169816730802, −7.30276772745260410694179205449, −6.30478645087209414468280431735, −5.67239272690558520293588017501, −5.18578726361425084528100652820, −4.50765887367555729721685928172, −3.61785217134958623400006565115, −2.65503796651228161044879027896, −1.52571952422545531473488459299, −0.852171786318819812294434282395, 0.852171786318819812294434282395, 1.52571952422545531473488459299, 2.65503796651228161044879027896, 3.61785217134958623400006565115, 4.50765887367555729721685928172, 5.18578726361425084528100652820, 5.67239272690558520293588017501, 6.30478645087209414468280431735, 7.30276772745260410694179205449, 7.87506370992720182169816730802

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