Properties

Label 2-2310-1.1-c3-0-20
Degree $2$
Conductor $2310$
Sign $1$
Analytic cond. $136.294$
Root an. cond. $11.6745$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s − 3·3-s + 4·4-s − 5·5-s + 6·6-s + 7·7-s − 8·8-s + 9·9-s + 10·10-s + 11·11-s − 12·12-s + 26·13-s − 14·14-s + 15·15-s + 16·16-s + 78·17-s − 18·18-s − 16·19-s − 20·20-s − 21·21-s − 22·22-s − 132·23-s + 24·24-s + 25·25-s − 52·26-s − 27·27-s + 28·28-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 + 1/3·9-s + 0.316·10-s + 0.301·11-s − 0.288·12-s + 0.554·13-s − 0.267·14-s + 0.258·15-s + 1/4·16-s + 1.11·17-s − 0.235·18-s − 0.193·19-s − 0.223·20-s − 0.218·21-s − 0.213·22-s − 1.19·23-s + 0.204·24-s + 1/5·25-s − 0.392·26-s − 0.192·27-s + 0.188·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2310 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2310 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2310\)    =    \(2 \cdot 3 \cdot 5 \cdot 7 \cdot 11\)
Sign: $1$
Analytic conductor: \(136.294\)
Root analytic conductor: \(11.6745\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2310,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(1.300839426\)
\(L(\frac12)\) \(\approx\) \(1.300839426\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + p T \)
3 \( 1 + p T \)
5 \( 1 + p T \)
7 \( 1 - p T \)
11 \( 1 - p T \)
good13 \( 1 - 2 p T + p^{3} T^{2} \)
17 \( 1 - 78 T + p^{3} T^{2} \)
19 \( 1 + 16 T + p^{3} T^{2} \)
23 \( 1 + 132 T + p^{3} T^{2} \)
29 \( 1 - 198 T + p^{3} T^{2} \)
31 \( 1 + 4 T + p^{3} T^{2} \)
37 \( 1 - 206 T + p^{3} T^{2} \)
41 \( 1 - 414 T + p^{3} T^{2} \)
43 \( 1 - 260 T + p^{3} T^{2} \)
47 \( 1 + 216 T + p^{3} T^{2} \)
53 \( 1 - 498 T + p^{3} T^{2} \)
59 \( 1 + 492 T + p^{3} T^{2} \)
61 \( 1 + 394 T + p^{3} T^{2} \)
67 \( 1 + 412 T + p^{3} T^{2} \)
71 \( 1 + 768 T + p^{3} T^{2} \)
73 \( 1 - 1238 T + p^{3} T^{2} \)
79 \( 1 - 560 T + p^{3} T^{2} \)
83 \( 1 + 900 T + p^{3} T^{2} \)
89 \( 1 + 954 T + p^{3} T^{2} \)
97 \( 1 - 182 T + p^{3} T^{2} \)
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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

−8.548083672409606020949920750481, −7.895479307081140811829176754552, −7.31049073557310887815098121813, −6.25128931037389208575144849721, −5.79502435814083546649375235898, −4.61378491810358526101745875074, −3.83957515815880860542394134421, −2.68691751875590113452347931317, −1.42769274913237279836088278561, −0.64208192913803307469393032837, 0.64208192913803307469393032837, 1.42769274913237279836088278561, 2.68691751875590113452347931317, 3.83957515815880860542394134421, 4.61378491810358526101745875074, 5.79502435814083546649375235898, 6.25128931037389208575144849721, 7.31049073557310887815098121813, 7.895479307081140811829176754552, 8.548083672409606020949920750481

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