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Properties

Label 2-303450-1.1-c1-0-121
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 − 12-s + 2·13-s + 14-s + 16-s + 18-s − 4·19-s − 21-s + 6·23-s − 24-s + 2·26-s − 27-s + 28-s − 5·29-s + 7·31-s + 32-s + 36-s + 5·37-s − 4·38-s − 2·39-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 − 0.288·12-s + 0.554·13-s + 0.267·14-s + 1/4·16-s + 0.235·18-s − 0.917·19-s − 0.218·21-s + 1.25·23-s − 0.204·24-s + 0.392·26-s − 0.192·27-s + 0.188·28-s − 0.928·29-s + 1.25·31-s + 0.176·32-s + 1/6·36-s + 0.821·37-s − 0.648·38-s − 0.320·39-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

\(L(1)\) \(\approx\) \(5.752627807\)
\(L(\frac12)\) \(\approx\) \(5.752627807\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 5 T + p T^{2} \)
31 \( 1 - 7 T + p T^{2} \)
37 \( 1 - 5 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 - 3 T + p T^{2} \)
53 \( 1 - 8 T + p T^{2} \)
59 \( 1 - 11 T + p T^{2} \)
61 \( 1 - 5 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 7 T + p T^{2} \)
79 \( 1 - 14 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 15 T + p T^{2} \)
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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

−12.75323571814328, −12.27399956529212, −11.70113368575290, −11.37135757400803, −10.91081406570169, −10.61782356948972, −10.15437106556112, −9.375599247569048, −9.099465347938490, −8.470880046510396, −7.893994515390702, −7.563038775336166, −6.800218107516257, −6.628173130666931, −6.002926892751760, −5.462366917420168, −5.266204580631056, −4.330425413640619, −4.288097078606164, −3.689522497360379, −2.907556282641032, −2.387194109599057, −1.895604093376859, −0.8581087150515279, −0.7923571970797910, 0.7923571970797910, 0.8581087150515279, 1.895604093376859, 2.387194109599057, 2.907556282641032, 3.689522497360379, 4.288097078606164, 4.330425413640619, 5.266204580631056, 5.462366917420168, 6.002926892751760, 6.628173130666931, 6.800218107516257, 7.563038775336166, 7.893994515390702, 8.470880046510396, 9.099465347938490, 9.375599247569048, 10.15437106556112, 10.61782356948972, 10.91081406570169, 11.37135757400803, 11.70113368575290, 12.27399956529212, 12.75323571814328

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