Properties

Label 2-40310-1.1-c1-0-19
Degree $2$
Conductor $40310$
Sign $1$
Analytic cond. $321.876$
Root an. cond. $17.9409$
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 + 5-s + 3·6-s + 7-s + 8-s + 6·9-s + 10-s − 2·11-s + 3·12-s + 5·13-s + 14-s + 3·15-s + 16-s + 3·17-s + 6·18-s + 5·19-s + 20-s + 3·21-s − 2·22-s + 2·23-s + 3·24-s + 25-s + 5·26-s + 9·27-s + 28-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.73·3-s + 1/2·4-s + 0.447·5-s + 1.22·6-s + 0.377·7-s + 0.353·8-s + 2·9-s + 0.316·10-s − 0.603·11-s + 0.866·12-s + 1.38·13-s + 0.267·14-s + 0.774·15-s + 1/4·16-s + 0.727·17-s + 1.41·18-s + 1.14·19-s + 0.223·20-s + 0.654·21-s − 0.426·22-s + 0.417·23-s + 0.612·24-s + 1/5·25-s + 0.980·26-s + 1.73·27-s + 0.188·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 40310 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40310 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(40310\)    =    \(2 \cdot 5 \cdot 29 \cdot 139\)
Sign: $1$
Analytic conductor: \(321.876\)
Root analytic conductor: \(17.9409\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 40310,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(11.98623458\)
\(L(\frac12)\) \(\approx\) \(11.98623458\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
5 \( 1 - T \)
29 \( 1 - T \)
139 \( 1 + T \)
good3 \( 1 - p T + p T^{2} \)
7 \( 1 - T + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
13 \( 1 - 5 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 - 5 T + p T^{2} \)
23 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 7 T + p T^{2} \)
67 \( 1 - 5 T + p T^{2} \)
71 \( 1 + 5 T + p T^{2} \)
73 \( 1 + 7 T + p T^{2} \)
79 \( 1 + 2 T + p T^{2} \)
83 \( 1 + 17 T + p T^{2} \)
89 \( 1 + 12 T + p T^{2} \)
97 \( 1 + 11 T + p T^{2} \)
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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

−14.60421113269213, −14.14404758940761, −13.83087454113131, −13.39089176450964, −12.95503274517302, −12.46485955693808, −11.71257531318242, −11.05886894655084, −10.62143726794184, −9.816829327166415, −9.546724910765800, −8.885657322932836, −8.190337672987343, −7.980175419055800, −7.351244126318028, −6.759825890370326, −5.890427648396441, −5.541732460877915, −4.610160443418596, −4.160738905758037, −3.423799224089570, −2.844313794393312, −2.590815627644945, −1.430989672564666, −1.223674065029220, 1.223674065029220, 1.430989672564666, 2.590815627644945, 2.844313794393312, 3.423799224089570, 4.160738905758037, 4.610160443418596, 5.541732460877915, 5.890427648396441, 6.759825890370326, 7.351244126318028, 7.980175419055800, 8.190337672987343, 8.885657322932836, 9.546724910765800, 9.816829327166415, 10.62143726794184, 11.05886894655084, 11.71257531318242, 12.46485955693808, 12.95503274517302, 13.39089176450964, 13.83087454113131, 14.14404758940761, 14.60421113269213

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