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Properties

Label 2-303450-1.1-c1-0-39
Degree $2$
Conductor $303450$
Sign $1$
Analytic cond. $2423.06$
Root an. cond. $49.2245$
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  + 2-s − 3-s + 4-s − 6-s − 7-s + 8-s + 9-s + 4·11-s − 12-s + 2·13-s − 14-s + 16-s + 18-s + 21-s + 4·22-s − 8·23-s − 24-s + 2·26-s − 27-s − 28-s − 10·29-s + 8·31-s + 32-s − 4·33-s + 36-s + 2·37-s − 2·39-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s + 1.20·11-s − 0.288·12-s + 0.554·13-s − 0.267·14-s + 1/4·16-s + 0.235·18-s + 0.218·21-s + 0.852·22-s − 1.66·23-s − 0.204·24-s + 0.392·26-s − 0.192·27-s − 0.188·28-s − 1.85·29-s + 1.43·31-s + 0.176·32-s − 0.696·33-s + 1/6·36-s + 0.328·37-s − 0.320·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 303450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 303450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(303450\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7 \cdot 17^{2}\)
Sign: $1$
Analytic conductor: \(2423.06\)
Root analytic conductor: \(49.2245\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 303450,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.110437434\)
\(L(\frac12)\) \(\approx\) \(3.110437434\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 + T \)
5 \( 1 \)
7 \( 1 + T \)
17 \( 1 \)
good11 \( 1 - 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 + 10 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
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

−12.67205866818347, −12.16776376832543, −11.74183048487389, −11.46952816561048, −11.07240842687450, −10.42163627388649, −9.974502156040074, −9.570478112639627, −9.126520311326275, −8.426826230198364, −7.969405149391916, −7.502274527110926, −6.796680091397685, −6.452214753222432, −6.107456180908263, −5.730993523440860, −5.020887265561644, −4.592889429993563, −3.927369802457071, −3.675350871955015, −3.193890162780535, −2.229514954322600, −1.850019660012788, −1.173632279657419, −0.4433681300794182, 0.4433681300794182, 1.173632279657419, 1.850019660012788, 2.229514954322600, 3.193890162780535, 3.675350871955015, 3.927369802457071, 4.592889429993563, 5.020887265561644, 5.730993523440860, 6.107456180908263, 6.452214753222432, 6.796680091397685, 7.502274527110926, 7.969405149391916, 8.426826230198364, 9.126520311326275, 9.570478112639627, 9.974502156040074, 10.42163627388649, 11.07240842687450, 11.46952816561048, 11.74183048487389, 12.16776376832543, 12.67205866818347

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