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Properties

Label 2-303450-1.1-c1-0-44
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 − 11-s − 12-s − 4·13-s + 14-s + 16-s − 18-s + 21-s + 22-s − 8·23-s + 24-s + 4·26-s − 27-s − 28-s − 29-s − 3·31-s − 32-s + 33-s + 36-s − 4·37-s + 4·39-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 − 0.301·11-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 + 0.213·22-s − 1.66·23-s + 0.204·24-s + 0.784·26-s − 0.192·27-s − 0.188·28-s − 0.185·29-s − 0.538·31-s − 0.176·32-s + 0.174·33-s + 1/6·36-s − 0.657·37-s + 0.640·39-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: \(1\)
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 \( 1 + T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 + T + p T^{2} \)
31 \( 1 + 3 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 + 4 T + p T^{2} \)
43 \( 1 - 6 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 5 T + p T^{2} \)
59 \( 1 + 5 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 5 T + p T^{2} \)
79 \( 1 + 3 T + p T^{2} \)
83 \( 1 + 16 T + p T^{2} \)
89 \( 1 + p T^{2} \)
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

−12.75685045287579, −12.40641616327051, −11.81218384790352, −11.67620098084481, −10.96996823578774, −10.57158933337263, −9.987928981789385, −9.898048563050129, −9.376426829451487, −8.678035501304641, −8.419835469641689, −7.569945932367700, −7.456610327285844, −6.982695883680383, −6.282253245107915, −5.943987983150018, −5.482068945285341, −4.857624519058523, −4.366135455355226, −3.738008280170678, −3.139102838287019, −2.499221254264067, −1.962312586743220, −1.438635922994569, −0.4720217479746260, 0, 0.4720217479746260, 1.438635922994569, 1.962312586743220, 2.499221254264067, 3.139102838287019, 3.738008280170678, 4.366135455355226, 4.857624519058523, 5.482068945285341, 5.943987983150018, 6.282253245107915, 6.982695883680383, 7.456610327285844, 7.569945932367700, 8.419835469641689, 8.678035501304641, 9.376426829451487, 9.898048563050129, 9.987928981789385, 10.57158933337263, 10.96996823578774, 11.67620098084481, 11.81218384790352, 12.40641616327051, 12.75685045287579

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