Properties

Label 2-7600-1.1-c1-0-2
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  − 1.77·3-s − 2.69·7-s + 0.144·9-s − 5.54·11-s + 2.91·13-s − 4.91·17-s − 19-s + 4.77·21-s + 3.60·23-s + 5.06·27-s + 1.08·29-s − 7.54·31-s + 9.83·33-s − 4.54·37-s − 5.17·39-s − 9.54·43-s − 0.836·47-s + 0.245·49-s + 8.72·51-s − 9.78·53-s + 1.77·57-s − 12.9·59-s − 7.38·61-s − 0.390·63-s + 2.85·67-s − 6.40·69-s − 14.4·71-s + ⋯
L(s)  = 1  − 1.02·3-s − 1.01·7-s + 0.0483·9-s − 1.67·11-s + 0.809·13-s − 1.19·17-s − 0.229·19-s + 1.04·21-s + 0.752·23-s + 0.974·27-s + 0.200·29-s − 1.35·31-s + 1.71·33-s − 0.747·37-s − 0.828·39-s − 1.45·43-s − 0.122·47-s + 0.0350·49-s + 1.22·51-s − 1.34·53-s + 0.234·57-s − 1.69·59-s − 0.945·61-s − 0.0491·63-s + 0.348·67-s − 0.770·69-s − 1.71·71-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: $\chi_{7600} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 7600,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1455139836\)
\(L(\frac12)\) \(\approx\) \(0.1455139836\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.77T + 3T^{2} \)
7 \( 1 + 2.69T + 7T^{2} \)
11 \( 1 + 5.54T + 11T^{2} \)
13 \( 1 - 2.91T + 13T^{2} \)
17 \( 1 + 4.91T + 17T^{2} \)
23 \( 1 - 3.60T + 23T^{2} \)
29 \( 1 - 1.08T + 29T^{2} \)
31 \( 1 + 7.54T + 31T^{2} \)
37 \( 1 + 4.54T + 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 + 9.54T + 43T^{2} \)
47 \( 1 + 0.836T + 47T^{2} \)
53 \( 1 + 9.78T + 53T^{2} \)
59 \( 1 + 12.9T + 59T^{2} \)
61 \( 1 + 7.38T + 61T^{2} \)
67 \( 1 - 2.85T + 67T^{2} \)
71 \( 1 + 14.4T + 71T^{2} \)
73 \( 1 + 5.15T + 73T^{2} \)
79 \( 1 - 3.09T + 79T^{2} \)
83 \( 1 - 1.71T + 83T^{2} \)
89 \( 1 + 5.09T + 89T^{2} \)
97 \( 1 - 17.2T + 97T^{2} \)
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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

−7.79174797032768961824426256385, −6.99199976176939709654568865396, −6.37268575609383337935529541308, −5.87777701792393756875776371624, −5.11409145155232521290687967199, −4.59014169564502552520265268804, −3.37455816351837369461855414315, −2.86037037760108248977006654470, −1.69996794284488110084518020994, −0.19245932654491665248069073239, 0.19245932654491665248069073239, 1.69996794284488110084518020994, 2.86037037760108248977006654470, 3.37455816351837369461855414315, 4.59014169564502552520265268804, 5.11409145155232521290687967199, 5.87777701792393756875776371624, 6.37268575609383337935529541308, 6.99199976176939709654568865396, 7.79174797032768961824426256385

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