Properties

Label 2-76664-1.1-c1-0-2
Degree $2$
Conductor $76664$
Sign $1$
Analytic cond. $612.165$
Root an. cond. $24.7419$
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  + 2·3-s − 5-s − 7-s + 9-s + 2·11-s + 2·13-s − 2·15-s − 17-s − 2·19-s − 2·21-s − 2·23-s − 4·25-s − 4·27-s + 3·29-s − 6·31-s + 4·33-s + 35-s + 4·39-s − 9·41-s + 8·43-s − 45-s + 6·47-s + 49-s − 2·51-s − 14·53-s − 2·55-s − 4·57-s + ⋯
L(s)  = 1  + 1.15·3-s − 0.447·5-s − 0.377·7-s + 1/3·9-s + 0.603·11-s + 0.554·13-s − 0.516·15-s − 0.242·17-s − 0.458·19-s − 0.436·21-s − 0.417·23-s − 4/5·25-s − 0.769·27-s + 0.557·29-s − 1.07·31-s + 0.696·33-s + 0.169·35-s + 0.640·39-s − 1.40·41-s + 1.21·43-s − 0.149·45-s + 0.875·47-s + 1/7·49-s − 0.280·51-s − 1.92·53-s − 0.269·55-s − 0.529·57-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76664 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76664 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76664\)    =    \(2^{3} \cdot 7 \cdot 37^{2}\)
Sign: $1$
Analytic conductor: \(612.165\)
Root analytic conductor: \(24.7419\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{76664} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 76664,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.287201698\)
\(L(\frac12)\) \(\approx\) \(2.287201698\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
37 \( 1 \)
good3 \( 1 - 2 T + p T^{2} \)
5 \( 1 + T + p T^{2} \)
11 \( 1 - 2 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + 2 T + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 9 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 + 14 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 5 T + p T^{2} \)
67 \( 1 - 14 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 - 14 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 + 17 T + p T^{2} \)
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

−13.98517114572129, −13.78497671603972, −13.13989451827769, −12.45349617623707, −12.30030563524695, −11.49112343923376, −11.00901827178120, −10.64222489676969, −9.710396127132862, −9.468777099947494, −8.995954922602928, −8.400301487702961, −7.992006167300076, −7.630377099193144, −6.737069356687052, −6.501456053977409, −5.755646426277198, −5.126976183354323, −4.257245668524163, −3.771557593253042, −3.510036209795931, −2.722776787983063, −2.108699849089473, −1.501803371347609, −0.4441665231591945, 0.4441665231591945, 1.501803371347609, 2.108699849089473, 2.722776787983063, 3.510036209795931, 3.771557593253042, 4.257245668524163, 5.126976183354323, 5.755646426277198, 6.501456053977409, 6.737069356687052, 7.630377099193144, 7.992006167300076, 8.400301487702961, 8.995954922602928, 9.468777099947494, 9.710396127132862, 10.64222489676969, 11.00901827178120, 11.49112343923376, 12.30030563524695, 12.45349617623707, 13.13989451827769, 13.78497671603972, 13.98517114572129

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