Properties

Label 2-2790-1.1-c1-0-17
Degree $2$
Conductor $2790$
Sign $1$
Analytic cond. $22.2782$
Root an. cond. $4.71998$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s − 5-s + 4.54·7-s − 8-s + 10-s + 2.54·11-s − 1.05·13-s − 4.54·14-s + 16-s + 4.39·17-s + 7.60·19-s − 20-s − 2.54·22-s + 1.20·23-s + 25-s + 1.05·26-s + 4.54·28-s − 6.10·29-s + 31-s − 32-s − 4.39·34-s − 4.54·35-s + 7.73·37-s − 7.60·38-s + 40-s − 6.10·41-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.5·4-s − 0.447·5-s + 1.71·7-s − 0.353·8-s + 0.316·10-s + 0.768·11-s − 0.291·13-s − 1.21·14-s + 0.250·16-s + 1.06·17-s + 1.74·19-s − 0.223·20-s − 0.543·22-s + 0.251·23-s + 0.200·25-s + 0.206·26-s + 0.859·28-s − 1.13·29-s + 0.179·31-s − 0.176·32-s − 0.753·34-s − 0.769·35-s + 1.27·37-s − 1.23·38-s + 0.158·40-s − 0.953·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2790 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2790 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2790\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 31\)
Sign: $1$
Analytic conductor: \(22.2782\)
Root analytic conductor: \(4.71998\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2790,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.722632232\)
\(L(\frac12)\) \(\approx\) \(1.722632232\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 \)
5 \( 1 + T \)
31 \( 1 - T \)
good7 \( 1 - 4.54T + 7T^{2} \)
11 \( 1 - 2.54T + 11T^{2} \)
13 \( 1 + 1.05T + 13T^{2} \)
17 \( 1 - 4.39T + 17T^{2} \)
19 \( 1 - 7.60T + 19T^{2} \)
23 \( 1 - 1.20T + 23T^{2} \)
29 \( 1 + 6.10T + 29T^{2} \)
37 \( 1 - 7.73T + 37T^{2} \)
41 \( 1 + 6.10T + 41T^{2} \)
43 \( 1 + 6.28T + 43T^{2} \)
47 \( 1 + 2.68T + 47T^{2} \)
53 \( 1 - 0.502T + 53T^{2} \)
59 \( 1 - 10.8T + 59T^{2} \)
61 \( 1 + 9.80T + 61T^{2} \)
67 \( 1 - 7.46T + 67T^{2} \)
71 \( 1 - 8.65T + 71T^{2} \)
73 \( 1 + 8.70T + 73T^{2} \)
79 \( 1 + 0.918T + 79T^{2} \)
83 \( 1 + 2.41T + 83T^{2} \)
89 \( 1 + 16.4T + 89T^{2} \)
97 \( 1 + 9.20T + 97T^{2} \)
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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

−8.704964740778554428564323147947, −7.947477590082471467212639198556, −7.59240548247915009182331370854, −6.83596306569965981690275784515, −5.59214028612187561373555415873, −5.04532554680886401281552173168, −4.01152825054695068992193717052, −3.04899269269836024182274431269, −1.72780972964094610765890792840, −1.00507496397471596218587421216, 1.00507496397471596218587421216, 1.72780972964094610765890792840, 3.04899269269836024182274431269, 4.01152825054695068992193717052, 5.04532554680886401281552173168, 5.59214028612187561373555415873, 6.83596306569965981690275784515, 7.59240548247915009182331370854, 7.947477590082471467212639198556, 8.704964740778554428564323147947

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