Properties

Label 2-303450-1.1-c1-0-83
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 − 5·11-s − 12-s + 5·13-s − 14-s + 16-s − 18-s − 4·19-s − 21-s + 5·22-s − 2·23-s + 24-s − 5·26-s − 27-s + 28-s + 4·29-s − 7·31-s − 32-s + 5·33-s + 36-s + 6·37-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 − 1.50·11-s − 0.288·12-s + 1.38·13-s − 0.267·14-s + 1/4·16-s − 0.235·18-s − 0.917·19-s − 0.218·21-s + 1.06·22-s − 0.417·23-s + 0.204·24-s − 0.980·26-s − 0.192·27-s + 0.188·28-s + 0.742·29-s − 1.25·31-s − 0.176·32-s + 0.870·33-s + 1/6·36-s + 0.986·37-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 \( 1 + 5 T + p T^{2} \)
13 \( 1 - 5 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 2 T + p T^{2} \)
29 \( 1 - 4 T + p T^{2} \)
31 \( 1 + 7 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 11 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 9 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 - 7 T + p T^{2} \)
71 \( 1 - 9 T + p T^{2} \)
73 \( 1 + 11 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 3 T + p T^{2} \)
89 \( 1 + 9 T + p T^{2} \)
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

−12.93350906630392, −12.45000090646453, −11.85704523804983, −11.35631039048875, −11.00237194166588, −10.60656352614230, −10.30209554819183, −9.862985324231636, −9.115410998439786, −8.733482426989391, −8.254097869587153, −7.918950561444204, −7.405340539619396, −6.860677711925380, −6.329442051914173, −5.902581742408535, −5.392050775750229, −5.028060710094356, −4.181296497517126, −3.912117089172150, −3.068602694207884, −2.531858968155327, −1.928367332769768, −1.368310054850286, −0.6498635503827738, 0, 0.6498635503827738, 1.368310054850286, 1.928367332769768, 2.531858968155327, 3.068602694207884, 3.912117089172150, 4.181296497517126, 5.028060710094356, 5.392050775750229, 5.902581742408535, 6.329442051914173, 6.860677711925380, 7.405340539619396, 7.918950561444204, 8.254097869587153, 8.733482426989391, 9.115410998439786, 9.862985324231636, 10.30209554819183, 10.60656352614230, 11.00237194166588, 11.35631039048875, 11.85704523804983, 12.45000090646453, 12.93350906630392

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