Properties

Label 2-40310-1.1-c1-0-20
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 − 3·7-s − 8-s − 2·9-s − 10-s − 2·11-s − 12-s − 5·13-s + 3·14-s − 15-s + 16-s + 17-s + 2·18-s − 19-s + 20-s + 3·21-s + 2·22-s − 4·23-s + 24-s + 25-s + 5·26-s + 5·27-s − 3·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s − 1.13·7-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 + 0.801·14-s − 0.258·15-s + 1/4·16-s + 0.242·17-s + 0.471·18-s − 0.229·19-s + 0.223·20-s + 0.654·21-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 − 0.566·28-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 \( 1 + 3 T + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
13 \( 1 + 5 T + p T^{2} \)
17 \( 1 - T + p T^{2} \)
19 \( 1 + T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
31 \( 1 - 10 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 7 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 11 T + p T^{2} \)
67 \( 1 + 7 T + p T^{2} \)
71 \( 1 - T + p T^{2} \)
73 \( 1 + 17 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 13 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 + 17 T + p T^{2} \)
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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

−15.44616759825014, −14.77723379798389, −14.38643073906321, −13.61131647258587, −13.24073115913288, −12.45664890032581, −12.17161744019683, −11.69648370088138, −11.02364714578287, −10.35328446049797, −9.989577058393511, −9.732297339500303, −9.002739382785286, −8.404056261732485, −7.868046051395565, −7.160799772298750, −6.689218301185839, −5.989254774698292, −5.785054631820648, −4.946061640972337, −4.386004439245018, −3.230613532242011, −2.797909435277998, −2.265719771203788, −1.227155572551433, 0, 0, 1.227155572551433, 2.265719771203788, 2.797909435277998, 3.230613532242011, 4.386004439245018, 4.946061640972337, 5.785054631820648, 5.989254774698292, 6.689218301185839, 7.160799772298750, 7.868046051395565, 8.404056261732485, 9.002739382785286, 9.732297339500303, 9.989577058393511, 10.35328446049797, 11.02364714578287, 11.69648370088138, 12.17161744019683, 12.45664890032581, 13.24073115913288, 13.61131647258587, 14.38643073906321, 14.77723379798389, 15.44616759825014

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