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Properties

Label 2-152460-1.1-c1-0-7
Degree $2$
Conductor $152460$
Sign $1$
Analytic cond. $1217.39$
Root an. cond. $34.8912$
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 + 7-s + 3·13-s − 17-s − 6·19-s − 6·23-s + 25-s − 9·29-s − 4·31-s + 35-s + 2·37-s − 4·41-s − 10·43-s + 47-s + 49-s − 4·53-s + 8·59-s + 8·61-s + 3·65-s + 12·67-s − 8·71-s − 2·73-s − 13·79-s − 4·83-s − 85-s − 4·89-s + 3·91-s + ⋯
L(s)  = 1  + 0.447·5-s + 0.377·7-s + 0.832·13-s − 0.242·17-s − 1.37·19-s − 1.25·23-s + 1/5·25-s − 1.67·29-s − 0.718·31-s + 0.169·35-s + 0.328·37-s − 0.624·41-s − 1.52·43-s + 0.145·47-s + 1/7·49-s − 0.549·53-s + 1.04·59-s + 1.02·61-s + 0.372·65-s + 1.46·67-s − 0.949·71-s − 0.234·73-s − 1.46·79-s − 0.439·83-s − 0.108·85-s − 0.423·89-s + 0.314·91-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 152460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152460 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(152460\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(1217.39\)
Root analytic conductor: \(34.8912\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 152460,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.353899093\)
\(L(\frac12)\) \(\approx\) \(1.353899093\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
7 \( 1 - T \)
11 \( 1 \)
good13 \( 1 - 3 T + p T^{2} \)
17 \( 1 + T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 9 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 4 T + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 - T + p T^{2} \)
53 \( 1 + 4 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 13 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 4 T + p T^{2} \)
97 \( 1 + 13 T + p T^{2} \)
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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

−13.29256620876000, −12.93008960451813, −12.52207515419228, −11.78292069464539, −11.32321603204867, −11.07005681220341, −10.42156923318756, −9.997661657105975, −9.561174511355226, −8.873583669893800, −8.462304233061408, −8.186795804823138, −7.445798899655714, −6.920921031375374, −6.404986788177575, −5.890960891105576, −5.479086505013869, −4.892988999521292, −4.092421423026503, −3.897484696998765, −3.181457741330398, −2.302726987348576, −1.885856888984701, −1.417765916026534, −0.3235861460803271, 0.3235861460803271, 1.417765916026534, 1.885856888984701, 2.302726987348576, 3.181457741330398, 3.897484696998765, 4.092421423026503, 4.892988999521292, 5.479086505013869, 5.890960891105576, 6.404986788177575, 6.920921031375374, 7.445798899655714, 8.186795804823138, 8.462304233061408, 8.873583669893800, 9.561174511355226, 9.997661657105975, 10.42156923318756, 11.07005681220341, 11.32321603204867, 11.78292069464539, 12.52207515419228, 12.93008960451813, 13.29256620876000

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