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Properties

Label 2-246202-1.1-c1-0-8
Degree $2$
Conductor $246202$
Sign $1$
Analytic cond. $1965.93$
Root an. cond. $44.3388$
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 − 3·3-s + 4-s + 4·5-s + 3·6-s − 7-s − 8-s + 6·9-s − 4·10-s + 11-s − 3·12-s + 13-s + 14-s − 12·15-s + 16-s − 7·17-s − 6·18-s + 4·20-s + 3·21-s − 22-s + 23-s + 3·24-s + 11·25-s − 26-s − 9·27-s − 28-s + 9·29-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.73·3-s + 1/2·4-s + 1.78·5-s + 1.22·6-s − 0.377·7-s − 0.353·8-s + 2·9-s − 1.26·10-s + 0.301·11-s − 0.866·12-s + 0.277·13-s + 0.267·14-s − 3.09·15-s + 1/4·16-s − 1.69·17-s − 1.41·18-s + 0.894·20-s + 0.654·21-s − 0.213·22-s + 0.208·23-s + 0.612·24-s + 11/5·25-s − 0.196·26-s − 1.73·27-s − 0.188·28-s + 1.67·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 246202 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 246202 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(246202\)    =    \(2 \cdot 11 \cdot 19^{2} \cdot 31\)
Sign: $1$
Analytic conductor: \(1965.93\)
Root analytic conductor: \(44.3388\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 246202,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.529769482\)
\(L(\frac12)\) \(\approx\) \(1.529769482\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
11 \( 1 - T \)
19 \( 1 \)
31 \( 1 + T \)
good3 \( 1 + p T + p T^{2} \)
5 \( 1 - 4 T + p T^{2} \)
7 \( 1 + T + p T^{2} \)
13 \( 1 - T + p T^{2} \)
17 \( 1 + 7 T + p T^{2} \)
23 \( 1 - T + p T^{2} \)
29 \( 1 - 9 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 - 10 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + T + p T^{2} \)
59 \( 1 - 9 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 5 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 7 T + p T^{2} \)
79 \( 1 - 2 T + p T^{2} \)
83 \( 1 - 8 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
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

−12.82163360994868, −12.37527897234479, −11.80479393922350, −11.37528017088329, −10.92488846193989, −10.41885946345286, −10.24172338337153, −9.796638719242039, −9.177991502162843, −8.873923064527479, −8.398048531183328, −7.449203989979008, −6.824592018588308, −6.633387430601080, −6.244117636570694, −5.926065938102495, −5.186707637122830, −4.996526566382265, −4.320365926514733, −3.590835941138369, −2.646411749336967, −2.205231643330002, −1.632265319927128, −0.9551056052724172, −0.5133313315417666, 0.5133313315417666, 0.9551056052724172, 1.632265319927128, 2.205231643330002, 2.646411749336967, 3.590835941138369, 4.320365926514733, 4.996526566382265, 5.186707637122830, 5.926065938102495, 6.244117636570694, 6.633387430601080, 6.824592018588308, 7.449203989979008, 8.398048531183328, 8.873923064527479, 9.177991502162843, 9.796638719242039, 10.24172338337153, 10.41885946345286, 10.92488846193989, 11.37528017088329, 11.80479393922350, 12.37527897234479, 12.82163360994868

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