Properties

Label 2-64715-1.1-c1-0-2
Degree $2$
Conductor $64715$
Sign $-1$
Analytic cond. $516.751$
Root an. cond. $22.7321$
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  − 3-s − 2·4-s + 5-s − 7-s − 2·9-s − 3·11-s + 2·12-s + 5·13-s − 15-s + 4·16-s + 3·17-s − 2·19-s − 2·20-s + 21-s − 6·23-s + 25-s + 5·27-s + 2·28-s − 3·29-s − 4·31-s + 3·33-s − 35-s + 4·36-s − 2·37-s − 5·39-s − 12·41-s + 6·44-s + ⋯
L(s)  = 1  − 0.577·3-s − 4-s + 0.447·5-s − 0.377·7-s − 2/3·9-s − 0.904·11-s + 0.577·12-s + 1.38·13-s − 0.258·15-s + 16-s + 0.727·17-s − 0.458·19-s − 0.447·20-s + 0.218·21-s − 1.25·23-s + 1/5·25-s + 0.962·27-s + 0.377·28-s − 0.557·29-s − 0.718·31-s + 0.522·33-s − 0.169·35-s + 2/3·36-s − 0.328·37-s − 0.800·39-s − 1.87·41-s + 0.904·44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 64715 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64715 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(64715\)    =    \(5 \cdot 7 \cdot 43^{2}\)
Sign: $-1$
Analytic conductor: \(516.751\)
Root analytic conductor: \(22.7321\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{64715} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 64715,\ (\ :1/2),\ -1)\)

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)$
bad5 \( 1 - T \)
7 \( 1 + T \)
43 \( 1 \)
good2 \( 1 + p T^{2} \)
3 \( 1 + T + p T^{2} \)
11 \( 1 + 3 T + p T^{2} \)
13 \( 1 - 5 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 12 T + p T^{2} \)
47 \( 1 - 9 T + p T^{2} \)
53 \( 1 - 12 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 12 T + p T^{2} \)
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

−14.42178425408011, −13.77106134627716, −13.41154403955116, −13.24592317262737, −12.30843983512943, −12.19334958969421, −11.47059735044065, −10.75229054108869, −10.35186940376949, −10.14634363325781, −9.234157248848498, −8.907781666521111, −8.370797614731858, −7.922897701276733, −7.241979448862465, −6.374963563052200, −5.957252729482368, −5.523089934493860, −5.168409759743318, −4.366185468954585, −3.638744415915559, −3.325929390910893, −2.393134444813448, −1.596175382999599, −0.6945055441275580, 0, 0.6945055441275580, 1.596175382999599, 2.393134444813448, 3.325929390910893, 3.638744415915559, 4.366185468954585, 5.168409759743318, 5.523089934493860, 5.957252729482368, 6.374963563052200, 7.241979448862465, 7.922897701276733, 8.370797614731858, 8.907781666521111, 9.234157248848498, 10.14634363325781, 10.35186940376949, 10.75229054108869, 11.47059735044065, 12.19334958969421, 12.30843983512943, 13.24592317262737, 13.41154403955116, 13.77106134627716, 14.42178425408011

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