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Properties

Label 2-60840-1.1-c1-0-23
Degree $2$
Conductor $60840$
Sign $1$
Analytic cond. $485.809$
Root an. cond. $22.0410$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5-s − 4·7-s + 4·11-s + 8·17-s − 6·23-s + 25-s + 8·29-s − 2·31-s − 4·35-s + 4·37-s − 10·41-s + 4·43-s − 8·47-s + 9·49-s + 10·53-s + 4·55-s + 12·59-s − 2·61-s + 2·67-s − 12·71-s + 2·73-s − 16·77-s + 8·79-s − 8·83-s + 8·85-s − 6·89-s + 10·97-s + ⋯
L(s)  = 1  + 0.447·5-s − 1.51·7-s + 1.20·11-s + 1.94·17-s − 1.25·23-s + 1/5·25-s + 1.48·29-s − 0.359·31-s − 0.676·35-s + 0.657·37-s − 1.56·41-s + 0.609·43-s − 1.16·47-s + 9/7·49-s + 1.37·53-s + 0.539·55-s + 1.56·59-s − 0.256·61-s + 0.244·67-s − 1.42·71-s + 0.234·73-s − 1.82·77-s + 0.900·79-s − 0.878·83-s + 0.867·85-s − 0.635·89-s + 1.01·97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60840\)    =    \(2^{3} \cdot 3^{2} \cdot 5 \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(485.809\)
Root analytic conductor: \(22.0410\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 60840,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.506246097\)
\(L(\frac12)\) \(\approx\) \(2.506246097\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 - T \)
13 \( 1 \)
good7 \( 1 + 4 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
17 \( 1 - 8 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 8 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 8 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−14.31827682268330, −13.75529959392353, −13.38843650532507, −12.65098837626722, −12.34659596848539, −11.84493725498160, −11.47329817351712, −10.42129166923081, −10.06022719034009, −9.824041301100555, −9.339548223380404, −8.598482661951749, −8.237875990252604, −7.359679844869202, −6.915472804319874, −6.326799811113852, −5.957173063839778, −5.462753876839657, −4.614963867912242, −3.867665009443891, −3.420918059312280, −2.929598945332072, −2.078079931708963, −1.253138520025687, −0.5833846364321820, 0.5833846364321820, 1.253138520025687, 2.078079931708963, 2.929598945332072, 3.420918059312280, 3.867665009443891, 4.614963867912242, 5.462753876839657, 5.957173063839778, 6.326799811113852, 6.915472804319874, 7.359679844869202, 8.237875990252604, 8.598482661951749, 9.339548223380404, 9.824041301100555, 10.06022719034009, 10.42129166923081, 11.47329817351712, 11.84493725498160, 12.34659596848539, 12.65098837626722, 13.38843650532507, 13.75529959392353, 14.31827682268330

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