Properties

Label 2-7600-1.1-c1-0-45
Degree $2$
Conductor $7600$
Sign $1$
Analytic cond. $60.6863$
Root an. cond. $7.79014$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 7-s − 2·9-s + 13-s + 3·17-s − 19-s − 21-s + 3·23-s − 5·27-s − 3·29-s − 2·31-s + 10·37-s + 39-s + 6·41-s + 2·43-s − 6·49-s + 3·51-s − 3·53-s − 57-s − 3·59-s + 8·61-s + 2·63-s − 7·67-s + 3·69-s − 12·71-s + 13·73-s − 14·79-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.377·7-s − 2/3·9-s + 0.277·13-s + 0.727·17-s − 0.229·19-s − 0.218·21-s + 0.625·23-s − 0.962·27-s − 0.557·29-s − 0.359·31-s + 1.64·37-s + 0.160·39-s + 0.937·41-s + 0.304·43-s − 6/7·49-s + 0.420·51-s − 0.412·53-s − 0.132·57-s − 0.390·59-s + 1.02·61-s + 0.251·63-s − 0.855·67-s + 0.361·69-s − 1.42·71-s + 1.52·73-s − 1.57·79-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: yes
Arithmetic: yes
Character: $\chi_{7600} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 7600,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.120760220\)
\(L(\frac12)\) \(\approx\) \(2.120760220\)
\(L(\frac{3}{2})\) 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 \( 1 - T + p T^{2} \)
7 \( 1 + T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 3 T + p T^{2} \)
59 \( 1 + 3 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 + 7 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 13 T + p T^{2} \)
79 \( 1 + 14 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−7.73936068719185659044756872795, −7.48462605502709942352829040599, −6.33139583952515351105997892284, −5.92973835035064815330717608734, −5.09504150507389868271007564322, −4.18943345205887335333213233881, −3.37192912202670177455074465984, −2.83437895172657956697323154633, −1.91468577956155137839871501691, −0.69699894662822146691634315191, 0.69699894662822146691634315191, 1.91468577956155137839871501691, 2.83437895172657956697323154633, 3.37192912202670177455074465984, 4.18943345205887335333213233881, 5.09504150507389868271007564322, 5.92973835035064815330717608734, 6.33139583952515351105997892284, 7.48462605502709942352829040599, 7.73936068719185659044756872795

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