Properties

Label 2-7600-1.1-c1-0-46
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

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 7-s − 2·9-s + 6·11-s − 5·13-s − 3·17-s − 19-s − 21-s + 3·23-s − 5·27-s + 9·29-s + 4·31-s + 6·33-s − 2·37-s − 5·39-s + 8·43-s − 6·49-s − 3·51-s + 3·53-s − 57-s − 9·59-s − 10·61-s + 2·63-s + 5·67-s + 3·69-s + 6·71-s + 7·73-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.377·7-s − 2/3·9-s + 1.80·11-s − 1.38·13-s − 0.727·17-s − 0.229·19-s − 0.218·21-s + 0.625·23-s − 0.962·27-s + 1.67·29-s + 0.718·31-s + 1.04·33-s − 0.328·37-s − 0.800·39-s + 1.21·43-s − 6/7·49-s − 0.420·51-s + 0.412·53-s − 0.132·57-s − 1.17·59-s − 1.28·61-s + 0.251·63-s + 0.610·67-s + 0.361·69-s + 0.712·71-s + 0.819·73-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: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 7600,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.151325934\)
\(L(\frac12)\) \(\approx\) \(2.151325934\)
\(L(\frac{3}{2})\) 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 \( 1 - 6 T + p T^{2} \)
13 \( 1 + 5 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 - 9 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 3 T + p T^{2} \)
59 \( 1 + 9 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 5 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 - 7 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 + 12 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.965660853998351532368982028444, −7.08971577668189492485989188394, −6.55874110484964269431451883562, −5.98727468337922828585960000348, −4.85431664116277716307335752712, −4.35446566947163584234256974578, −3.39515826505100747996509869374, −2.75515250851414006834070112474, −1.94400683778762803472021635957, −0.70045234573102504240688012324, 0.70045234573102504240688012324, 1.94400683778762803472021635957, 2.75515250851414006834070112474, 3.39515826505100747996509869374, 4.35446566947163584234256974578, 4.85431664116277716307335752712, 5.98727468337922828585960000348, 6.55874110484964269431451883562, 7.08971577668189492485989188394, 7.965660853998351532368982028444

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