Properties

Label 2-344760-1.1-c1-0-32
Degree $2$
Conductor $344760$
Sign $-1$
Analytic cond. $2752.92$
Root an. cond. $52.4682$
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 + 5-s + 2·7-s + 9-s − 5·11-s − 15-s + 17-s − 6·19-s − 2·21-s − 3·23-s + 25-s − 27-s + 3·29-s + 7·31-s + 5·33-s + 2·35-s + 5·37-s + 2·41-s + 43-s + 45-s − 7·47-s − 3·49-s − 51-s + 6·53-s − 5·55-s + 6·57-s − 11·59-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447·5-s + 0.755·7-s + 1/3·9-s − 1.50·11-s − 0.258·15-s + 0.242·17-s − 1.37·19-s − 0.436·21-s − 0.625·23-s + 1/5·25-s − 0.192·27-s + 0.557·29-s + 1.25·31-s + 0.870·33-s + 0.338·35-s + 0.821·37-s + 0.312·41-s + 0.152·43-s + 0.149·45-s − 1.02·47-s − 3/7·49-s − 0.140·51-s + 0.824·53-s − 0.674·55-s + 0.794·57-s − 1.43·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 344760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 344760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(344760\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 13^{2} \cdot 17\)
Sign: $-1$
Analytic conductor: \(2752.92\)
Root analytic conductor: \(52.4682\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 344760,\ (\ :1/2),\ -1)\)

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 \( 1 \)
3 \( 1 + T \)
5 \( 1 - T \)
13 \( 1 \)
17 \( 1 - T \)
good7 \( 1 - 2 T + p T^{2} \)
11 \( 1 + 5 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 - 7 T + p T^{2} \)
37 \( 1 - 5 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 + 7 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 11 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 - 6 T + p T^{2} \)
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

−12.79923540156747, −12.41430937471182, −11.73200242544846, −11.51923881625728, −10.86825335485148, −10.50145990732790, −10.23772777703043, −9.777320982340693, −9.186273855642017, −8.496093175948793, −8.180241189831203, −7.832370766032975, −7.302830821005061, −6.611232858573968, −6.238040888515778, −5.778038756436961, −5.290416861544722, −4.747759053728816, −4.474077549876928, −3.907161605576536, −2.915492547551112, −2.663608817972092, −1.971288225661149, −1.484354356358838, −0.6863412781763340, 0, 0.6863412781763340, 1.484354356358838, 1.971288225661149, 2.663608817972092, 2.915492547551112, 3.907161605576536, 4.474077549876928, 4.747759053728816, 5.290416861544722, 5.778038756436961, 6.238040888515778, 6.611232858573968, 7.302830821005061, 7.832370766032975, 8.180241189831203, 8.496093175948793, 9.186273855642017, 9.777320982340693, 10.23772777703043, 10.50145990732790, 10.86825335485148, 11.51923881625728, 11.73200242544846, 12.41430937471182, 12.79923540156747

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