Properties

Degree $2$
Conductor $248430$
Sign $1$
Motivic weight $1$
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 + 5-s − 6-s + 8-s + 9-s + 10-s − 12-s − 15-s + 16-s + 6·17-s + 18-s + 8·19-s + 20-s − 24-s + 25-s − 27-s + 6·29-s − 30-s − 4·31-s + 32-s + 6·34-s + 36-s + 10·37-s + 8·38-s + 40-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s + 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.288·12-s − 0.258·15-s + 1/4·16-s + 1.45·17-s + 0.235·18-s + 1.83·19-s + 0.223·20-s − 0.204·24-s + 1/5·25-s − 0.192·27-s + 1.11·29-s − 0.182·30-s − 0.718·31-s + 0.176·32-s + 1.02·34-s + 1/6·36-s + 1.64·37-s + 1.29·38-s + 0.158·40-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 248430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 248430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(248430\)    =    \(2 \cdot 3 \cdot 5 \cdot 7^{2} \cdot 13^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{248430} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 248430,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.086614826\)
\(L(\frac12)\) \(\approx\) \(5.086614826\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
7 \( 1 \)
13 \( 1 \)
good11 \( 1 + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−12.90239056195395, −12.20712472077785, −12.04588090537994, −11.68106491018336, −10.99728149641252, −10.73347131104802, −9.967423002967585, −9.771176139317661, −9.409548936651184, −8.572797786074299, −8.021138748396791, −7.534141938844030, −7.177012153785536, −6.519595736830906, −6.085993355318617, −5.614398163148114, −5.183881923679723, −4.839169250461871, −4.174156697571255, −3.492052373680013, −3.069147288141990, −2.607572177068305, −1.616052581485161, −1.279424933186272, −0.5929825374163094, 0.5929825374163094, 1.279424933186272, 1.616052581485161, 2.607572177068305, 3.069147288141990, 3.492052373680013, 4.174156697571255, 4.839169250461871, 5.183881923679723, 5.614398163148114, 6.085993355318617, 6.519595736830906, 7.177012153785536, 7.534141938844030, 8.021138748396791, 8.572797786074299, 9.409548936651184, 9.771176139317661, 9.967423002967585, 10.73347131104802, 10.99728149641252, 11.68106491018336, 12.04588090537994, 12.20712472077785, 12.90239056195395

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