Properties

Label 2-303450-1.1-c1-0-10
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 − 6·11-s + 12-s + 13-s + 14-s + 16-s + 18-s − 4·19-s + 21-s − 6·22-s − 3·23-s + 24-s + 26-s + 27-s + 28-s − 3·29-s − 5·31-s + 32-s − 6·33-s + 36-s − 10·37-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 − 1.80·11-s + 0.288·12-s + 0.277·13-s + 0.267·14-s + 1/4·16-s + 0.235·18-s − 0.917·19-s + 0.218·21-s − 1.27·22-s − 0.625·23-s + 0.204·24-s + 0.196·26-s + 0.192·27-s + 0.188·28-s − 0.557·29-s − 0.898·31-s + 0.176·32-s − 1.04·33-s + 1/6·36-s − 1.64·37-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

\(L(1)\) \(\approx\) \(1.609229260\)
\(L(\frac12)\) \(\approx\) \(1.609229260\)
\(L(\frac{3}{2})\) 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 \( 1 + 6 T + p T^{2} \)
13 \( 1 - T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 + 5 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + 9 T + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 - 9 T + p T^{2} \)
61 \( 1 + 11 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 + 9 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
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

−12.74545694840307, −12.36590701192696, −11.92804430436805, −11.19435402051744, −10.85844031987531, −10.50407212904580, −10.05777785047196, −9.526191961465698, −8.867716270601296, −8.319331174338111, −8.132707329011851, −7.577432280796870, −7.079627688835201, −6.650972087991846, −5.945249604294631, −5.466703949580421, −5.095138688959806, −4.589360025037038, −4.006098191863684, −3.430708081497057, −3.069774131101353, −2.291584859058262, −2.000369582144807, −1.441757788612557, −0.2549111807672670, 0.2549111807672670, 1.441757788612557, 2.000369582144807, 2.291584859058262, 3.069774131101353, 3.430708081497057, 4.006098191863684, 4.589360025037038, 5.095138688959806, 5.466703949580421, 5.945249604294631, 6.650972087991846, 7.079627688835201, 7.577432280796870, 8.132707329011851, 8.319331174338111, 8.867716270601296, 9.526191961465698, 10.05777785047196, 10.50407212904580, 10.85844031987531, 11.19435402051744, 11.92804430436805, 12.36590701192696, 12.74545694840307

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