Properties

Label 2-303450-1.1-c1-0-102
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 + 2·11-s + 12-s + 4·13-s + 14-s + 16-s + 18-s + 21-s + 2·22-s + 8·23-s + 24-s + 4·26-s + 27-s + 28-s + 4·31-s + 32-s + 2·33-s + 36-s − 2·37-s + 4·39-s − 6·41-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 + 0.603·11-s + 0.288·12-s + 1.10·13-s + 0.267·14-s + 1/4·16-s + 0.235·18-s + 0.218·21-s + 0.426·22-s + 1.66·23-s + 0.204·24-s + 0.784·26-s + 0.192·27-s + 0.188·28-s + 0.718·31-s + 0.176·32-s + 0.348·33-s + 1/6·36-s − 0.328·37-s + 0.640·39-s − 0.937·41-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

\(L(1)\) \(\approx\) \(9.041403735\)
\(L(\frac12)\) \(\approx\) \(9.041403735\)
\(L(\frac{3}{2})\) 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 \( 1 - 2 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 - 2 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 10 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 + 14 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 + 18 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−12.80719835473452, −12.26527949479244, −11.79782542794588, −11.33303505472530, −10.93410796016868, −10.55483935784957, −9.917357044889915, −9.443050913747678, −8.884882908856821, −8.474637663383799, −8.190782019604837, −7.445745598713163, −7.005408819717359, −6.601371651454347, −6.156259300817625, −5.395298656154335, −5.159144350232351, −4.405582024321593, −4.060693568188953, −3.493899505868956, −3.032314050352541, −2.521160937356467, −1.742252302060672, −1.301313691174573, −0.7028077255428153, 0.7028077255428153, 1.301313691174573, 1.742252302060672, 2.521160937356467, 3.032314050352541, 3.493899505868956, 4.060693568188953, 4.405582024321593, 5.159144350232351, 5.395298656154335, 6.156259300817625, 6.601371651454347, 7.005408819717359, 7.445745598713163, 8.190782019604837, 8.474637663383799, 8.884882908856821, 9.443050913747678, 9.917357044889915, 10.55483935784957, 10.93410796016868, 11.33303505472530, 11.79782542794588, 12.26527949479244, 12.80719835473452

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