Properties

Label 2-303450-1.1-c1-0-86
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 $1$

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 + 6·19-s + 21-s − 2·22-s + 4·23-s + 24-s + 4·26-s − 27-s − 28-s − 7·29-s + 3·31-s − 32-s − 2·33-s + 36-s + 5·37-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 + 0.603·11-s − 0.288·12-s − 1.10·13-s + 0.267·14-s + 1/4·16-s − 0.235·18-s + 1.37·19-s + 0.218·21-s − 0.426·22-s + 0.834·23-s + 0.204·24-s + 0.784·26-s − 0.192·27-s − 0.188·28-s − 1.29·29-s + 0.538·31-s − 0.176·32-s − 0.348·33-s + 1/6·36-s + 0.821·37-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: $\chi_{303450} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 303450,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 - 6 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 7 T + p T^{2} \)
31 \( 1 - 3 T + p T^{2} \)
37 \( 1 - 5 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 + 3 T + p T^{2} \)
53 \( 1 + 14 T + p T^{2} \)
59 \( 1 - T + p T^{2} \)
61 \( 1 + 15 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 11 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 8 T + p T^{2} \)
89 \( 1 - 18 T + p T^{2} \)
97 \( 1 - 9 T + p T^{2} \)
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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

−12.78183797184099, −12.33635027673251, −11.78857583437991, −11.60487490254271, −11.12385908096865, −10.56763708620053, −10.09074721890564, −9.633508768566281, −9.294536972213184, −9.051057961963749, −8.184200040227624, −7.694195252654120, −7.409038805566161, −6.865467786030356, −6.418775105518735, −5.967845201039372, −5.379470650281537, −4.814786221292758, −4.526808532415399, −3.519887773287744, −3.248089086411039, −2.633148048714286, −1.826765565701623, −1.375626716256705, −0.6517697381802282, 0, 0.6517697381802282, 1.375626716256705, 1.826765565701623, 2.633148048714286, 3.248089086411039, 3.519887773287744, 4.526808532415399, 4.814786221292758, 5.379470650281537, 5.967845201039372, 6.418775105518735, 6.865467786030356, 7.409038805566161, 7.694195252654120, 8.184200040227624, 9.051057961963749, 9.294536972213184, 9.633508768566281, 10.09074721890564, 10.56763708620053, 11.12385908096865, 11.60487490254271, 11.78857583437991, 12.33635027673251, 12.78183797184099

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