Properties

Label 2-248430-1.1-c1-0-122
Degree $2$
Conductor $248430$
Sign $1$
Analytic cond. $1983.72$
Root an. cond. $44.5390$
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 − 5-s + 6-s + 8-s + 9-s − 10-s + 4·11-s + 12-s − 15-s + 16-s + 6·17-s + 18-s − 20-s + 4·22-s − 8·23-s + 24-s + 25-s + 27-s + 10·29-s − 30-s − 8·31-s + 32-s + 4·33-s + 6·34-s + 36-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 + 1.20·11-s + 0.288·12-s − 0.258·15-s + 1/4·16-s + 1.45·17-s + 0.235·18-s − 0.223·20-s + 0.852·22-s − 1.66·23-s + 0.204·24-s + 1/5·25-s + 0.192·27-s + 1.85·29-s − 0.182·30-s − 1.43·31-s + 0.176·32-s + 0.696·33-s + 1.02·34-s + 1/6·36-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: $1$
Analytic conductor: \(1983.72\)
Root analytic conductor: \(44.5390\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{248430} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 248430,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(7.285848739\)
\(L(\frac12)\) \(\approx\) \(7.285848739\)
\(L(\frac{3}{2})\) 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 \( 1 - 4 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 - 10 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−12.72373262850118, −12.40222345160165, −11.96285854567205, −11.75173008722610, −11.14463247756419, −10.46478863264330, −10.11421327843777, −9.711678923858693, −9.060137347233257, −8.545321844404977, −8.229812259152852, −7.521379516186666, −7.261411022820034, −6.743347455855132, −5.980377675882019, −5.859768061174512, −5.069272004978112, −4.507014732864872, −3.953138097688228, −3.630259190961427, −3.235344116319972, −2.407390676567886, −1.981267276012844, −1.182069315377889, −0.6761045295338733, 0.6761045295338733, 1.182069315377889, 1.981267276012844, 2.407390676567886, 3.235344116319972, 3.630259190961427, 3.953138097688228, 4.507014732864872, 5.069272004978112, 5.859768061174512, 5.980377675882019, 6.743347455855132, 7.261411022820034, 7.521379516186666, 8.229812259152852, 8.545321844404977, 9.060137347233257, 9.711678923858693, 10.11421327843777, 10.46478863264330, 11.14463247756419, 11.75173008722610, 11.96285854567205, 12.40222345160165, 12.72373262850118

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