Properties

Label 2-152460-1.1-c1-0-29
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 + 6·13-s + 2·17-s − 6·23-s + 25-s + 6·29-s − 2·31-s + 35-s + 10·37-s − 8·41-s + 8·43-s − 4·47-s + 49-s − 6·53-s + 6·59-s + 8·61-s + 6·65-s + 14·67-s + 8·71-s + 2·73-s + 16·79-s + 12·83-s + 2·85-s + 18·89-s + 6·91-s + 2·97-s + ⋯
L(s)  = 1  + 0.447·5-s + 0.377·7-s + 1.66·13-s + 0.485·17-s − 1.25·23-s + 1/5·25-s + 1.11·29-s − 0.359·31-s + 0.169·35-s + 1.64·37-s − 1.24·41-s + 1.21·43-s − 0.583·47-s + 1/7·49-s − 0.824·53-s + 0.781·59-s + 1.02·61-s + 0.744·65-s + 1.71·67-s + 0.949·71-s + 0.234·73-s + 1.80·79-s + 1.31·83-s + 0.216·85-s + 1.90·89-s + 0.628·91-s + 0.203·97-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: $\chi_{152460} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 152460,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.362953802\)
\(L(\frac12)\) \(\approx\) \(4.362953802\)
\(L(\frac{3}{2})\) 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 \( 1 - 6 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 8 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 - 14 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 18 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−13.43200378704735, −12.84063211741602, −12.46474402037146, −11.77410809781253, −11.49428074112246, −10.81312326569174, −10.61792921554599, −9.917398416744726, −9.531328340318483, −9.026199989186618, −8.291888941463955, −8.133026269503353, −7.682397579029747, −6.693956597801202, −6.487529796303927, −5.951563698259819, −5.428965954951590, −4.906365913347291, −4.218348605332027, −3.687285355748632, −3.265128232600726, −2.328332274941515, −1.972965969009865, −1.096952880393825, −0.7093963255708280, 0.7093963255708280, 1.096952880393825, 1.972965969009865, 2.328332274941515, 3.265128232600726, 3.687285355748632, 4.218348605332027, 4.906365913347291, 5.428965954951590, 5.951563698259819, 6.487529796303927, 6.693956597801202, 7.682397579029747, 8.133026269503353, 8.291888941463955, 9.026199989186618, 9.531328340318483, 9.917398416744726, 10.61792921554599, 10.81312326569174, 11.49428074112246, 11.77410809781253, 12.46474402037146, 12.84063211741602, 13.43200378704735

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