Properties

Label 2-75712-1.1-c1-0-39
Degree $2$
Conductor $75712$
Sign $-1$
Analytic cond. $604.563$
Root an. cond. $24.5878$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s − 3·5-s + 7-s + 9-s + 6·15-s − 6·17-s + 7·19-s − 2·21-s − 3·23-s + 4·25-s + 4·27-s + 9·29-s + 5·31-s − 3·35-s + 2·37-s + 6·41-s − 43-s − 3·45-s + 3·47-s + 49-s + 12·51-s + 9·53-s − 14·57-s + 10·61-s + 63-s − 14·67-s + 6·69-s + ⋯
L(s)  = 1  − 1.15·3-s − 1.34·5-s + 0.377·7-s + 1/3·9-s + 1.54·15-s − 1.45·17-s + 1.60·19-s − 0.436·21-s − 0.625·23-s + 4/5·25-s + 0.769·27-s + 1.67·29-s + 0.898·31-s − 0.507·35-s + 0.328·37-s + 0.937·41-s − 0.152·43-s − 0.447·45-s + 0.437·47-s + 1/7·49-s + 1.68·51-s + 1.23·53-s − 1.85·57-s + 1.28·61-s + 0.125·63-s − 1.71·67-s + 0.722·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75712 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(75712\)    =    \(2^{6} \cdot 7 \cdot 13^{2}\)
Sign: $-1$
Analytic conductor: \(604.563\)
Root analytic conductor: \(24.5878\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{75712} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 75712,\ (\ :1/2),\ -1)\)

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 \( 1 \)
7 \( 1 - T \)
13 \( 1 \)
good3 \( 1 + 2 T + p T^{2} \)
5 \( 1 + 3 T + p T^{2} \)
11 \( 1 + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 7 T + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 - 9 T + p T^{2} \)
31 \( 1 - 5 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + T + p T^{2} \)
47 \( 1 - 3 T + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 14 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 + 11 T + p T^{2} \)
79 \( 1 - T + p T^{2} \)
83 \( 1 + 3 T + p T^{2} \)
89 \( 1 + 15 T + p T^{2} \)
97 \( 1 - T + p T^{2} \)
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

−14.29410559654615, −13.75281970024720, −13.34451844197946, −12.52667398336632, −12.09589855459471, −11.72496536637971, −11.45684694979608, −11.03968759510621, −10.34367125359209, −10.06471934173203, −9.150128918073006, −8.623745242399577, −8.186373243893771, −7.581673969607055, −7.093627645390466, −6.597580227940029, −5.968620547896759, −5.439706750582827, −4.771893158377331, −4.354922239251239, −3.951981838555965, −2.950508031347986, −2.561080392942125, −1.324836742072009, −0.7366024105076901, 0, 0.7366024105076901, 1.324836742072009, 2.561080392942125, 2.950508031347986, 3.951981838555965, 4.354922239251239, 4.771893158377331, 5.439706750582827, 5.968620547896759, 6.597580227940029, 7.093627645390466, 7.581673969607055, 8.186373243893771, 8.623745242399577, 9.150128918073006, 10.06471934173203, 10.34367125359209, 11.03968759510621, 11.45684694979608, 11.72496536637971, 12.09589855459471, 12.52667398336632, 13.34451844197946, 13.75281970024720, 14.29410559654615

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