Properties

Label 2-40310-1.1-c1-0-26
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 $2$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3-s + 4-s + 5-s + 6-s + 7-s − 8-s − 2·9-s − 10-s − 2·11-s − 12-s − 5·13-s − 14-s − 15-s + 16-s − 3·17-s + 2·18-s − 5·19-s + 20-s − 21-s + 2·22-s + 4·23-s + 24-s + 25-s + 5·26-s + 5·27-s + 28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s + 0.377·7-s − 0.353·8-s − 2/3·9-s − 0.316·10-s − 0.603·11-s − 0.288·12-s − 1.38·13-s − 0.267·14-s − 0.258·15-s + 1/4·16-s − 0.727·17-s + 0.471·18-s − 1.14·19-s + 0.223·20-s − 0.218·21-s + 0.426·22-s + 0.834·23-s + 0.204·24-s + 1/5·25-s + 0.980·26-s + 0.962·27-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: \(2\)
Selberg data: \((2,\ 40310,\ (\ :1/2),\ 1)\)

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 \( 1 + T \)
5 \( 1 - T \)
29 \( 1 + T \)
139 \( 1 + T \)
good3 \( 1 + T + p T^{2} \)
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 \( 1 - 4 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + 7 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 7 T + p T^{2} \)
67 \( 1 - T + p T^{2} \)
71 \( 1 - T + p T^{2} \)
73 \( 1 + 5 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 - 11 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
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

−15.24514138251980, −14.93872722012921, −14.34629210747548, −13.73566262946394, −13.19184225633638, −12.50101529000292, −12.13474852888660, −11.55689850169186, −10.98072361624459, −10.57299564139429, −10.14939297342147, −9.537144575277013, −8.763563356101776, −8.590954456612275, −7.862612099046117, −7.209328401120816, −6.635100199505160, −6.269041730255042, −5.278081431953229, −5.124000656759516, −4.478672559876515, −3.355461428235752, −2.664088493071863, −2.127254126686609, −1.397463004086718, 0, 0, 1.397463004086718, 2.127254126686609, 2.664088493071863, 3.355461428235752, 4.478672559876515, 5.124000656759516, 5.278081431953229, 6.269041730255042, 6.635100199505160, 7.209328401120816, 7.862612099046117, 8.590954456612275, 8.763563356101776, 9.537144575277013, 10.14939297342147, 10.57299564139429, 10.98072361624459, 11.55689850169186, 12.13474852888660, 12.50101529000292, 13.19184225633638, 13.73566262946394, 14.34629210747548, 14.93872722012921, 15.24514138251980

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