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Properties

Label 2-40310-1.1-c1-0-7
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-s + 4-s − 5-s + 6-s + 5·7-s + 8-s − 2·9-s − 10-s − 6·11-s + 12-s + 5·13-s + 5·14-s − 15-s + 16-s − 3·17-s − 2·18-s − 19-s − 20-s + 5·21-s − 6·22-s + 6·23-s + 24-s + 25-s + 5·26-s − 5·27-s + 5·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s + 1.88·7-s + 0.353·8-s − 2/3·9-s − 0.316·10-s − 1.80·11-s + 0.288·12-s + 1.38·13-s + 1.33·14-s − 0.258·15-s + 1/4·16-s − 0.727·17-s − 0.471·18-s − 0.229·19-s − 0.223·20-s + 1.09·21-s − 1.27·22-s + 1.25·23-s + 0.204·24-s + 1/5·25-s + 0.980·26-s − 0.962·27-s + 0.944·28-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

\(L(1)\) \(\approx\) \(5.049961571\)
\(L(\frac12)\) \(\approx\) \(5.049961571\)
\(L(\frac{3}{2})\) 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 \( 1 - 5 T + p T^{2} \)
11 \( 1 + 6 T + p T^{2} \)
13 \( 1 - 5 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 + T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
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 \( 1 - 12 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 13 T + p T^{2} \)
67 \( 1 - 5 T + p T^{2} \)
71 \( 1 - 3 T + p T^{2} \)
73 \( 1 - 11 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 - 3 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 + T + p T^{2} \)
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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

−14.96372785152562, −14.15896832336359, −13.74364637313069, −13.40778787321319, −12.83865743970018, −12.24009862713436, −11.40310304950896, −11.18244649505932, −10.83530984840432, −10.41781207254362, −9.253420073756281, −8.662662429538813, −8.342799931307725, −7.829896050258661, −7.467080693708797, −6.684419965805504, −5.721528653069111, −5.471708472739204, −4.763687899415500, −4.348368179549606, −3.575190122424362, −2.870613257891162, −2.341074911512905, −1.667505494455712, −0.6941483256393399, 0.6941483256393399, 1.667505494455712, 2.341074911512905, 2.870613257891162, 3.575190122424362, 4.348368179549606, 4.763687899415500, 5.471708472739204, 5.721528653069111, 6.684419965805504, 7.467080693708797, 7.829896050258661, 8.342799931307725, 8.662662429538813, 9.253420073756281, 10.41781207254362, 10.83530984840432, 11.18244649505932, 11.40310304950896, 12.24009862713436, 12.83865743970018, 13.40778787321319, 13.74364637313069, 14.15896832336359, 14.96372785152562

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