Properties

Label 2-303450-1.1-c1-0-8
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 − 5·11-s − 12-s − 14-s + 16-s − 18-s + 6·19-s − 21-s + 5·22-s − 2·23-s + 24-s − 27-s + 28-s + 9·29-s + 3·31-s − 32-s + 5·33-s + 36-s + 6·37-s − 6·38-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 − 1.50·11-s − 0.288·12-s − 0.267·14-s + 1/4·16-s − 0.235·18-s + 1.37·19-s − 0.218·21-s + 1.06·22-s − 0.417·23-s + 0.204·24-s − 0.192·27-s + 0.188·28-s + 1.67·29-s + 0.538·31-s − 0.176·32-s + 0.870·33-s + 1/6·36-s + 0.986·37-s − 0.973·38-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: $\chi_{303450} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 303450,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5293426572\)
\(L(\frac12)\) \(\approx\) \(0.5293426572\)
\(L(\frac{3}{2})\) 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 \( 1 + 5 T + p T^{2} \)
13 \( 1 + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 + 2 T + p T^{2} \)
29 \( 1 - 9 T + p T^{2} \)
31 \( 1 - 3 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 + 3 T + p T^{2} \)
59 \( 1 + 13 T + p T^{2} \)
61 \( 1 + 4 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 + T + p T^{2} \)
79 \( 1 + 13 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 4 T + p T^{2} \)
97 \( 1 - 5 T + p T^{2} \)
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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

−12.35256059796003, −12.28164581399132, −11.61439332202765, −11.31501827253602, −10.79828941170487, −10.30655289551857, −10.01959322175077, −9.654137838022677, −8.961228013754816, −8.450244456474049, −8.013112362154225, −7.576809567846645, −7.304945561421312, −6.580094689521844, −6.089402695569999, −5.682865774522575, −5.067796297307977, −4.714936929345359, −4.206316275871097, −3.176413383225794, −2.918055778172830, −2.365439476910022, −1.453078348402021, −1.157398585429188, −0.2417239065160281, 0.2417239065160281, 1.157398585429188, 1.453078348402021, 2.365439476910022, 2.918055778172830, 3.176413383225794, 4.206316275871097, 4.714936929345359, 5.067796297307977, 5.682865774522575, 6.089402695569999, 6.580094689521844, 7.304945561421312, 7.576809567846645, 8.013112362154225, 8.450244456474049, 8.961228013754816, 9.654137838022677, 10.01959322175077, 10.30655289551857, 10.79828941170487, 11.31501827253602, 11.61439332202765, 12.28164581399132, 12.35256059796003

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