Properties

Label 2-5577-1.1-c1-0-71
Degree $2$
Conductor $5577$
Sign $1$
Analytic cond. $44.5325$
Root an. cond. $6.67327$
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  − 2.15·2-s + 3-s + 2.65·4-s + 0.710·5-s − 2.15·6-s + 2.30·7-s − 1.41·8-s + 9-s − 1.53·10-s + 11-s + 2.65·12-s − 4.98·14-s + 0.710·15-s − 2.26·16-s − 6.68·17-s − 2.15·18-s − 0.242·19-s + 1.88·20-s + 2.30·21-s − 2.15·22-s + 9.53·23-s − 1.41·24-s − 4.49·25-s + 27-s + 6.13·28-s − 2.95·29-s − 1.53·30-s + ⋯
L(s)  = 1  − 1.52·2-s + 0.577·3-s + 1.32·4-s + 0.317·5-s − 0.880·6-s + 0.872·7-s − 0.499·8-s + 0.333·9-s − 0.484·10-s + 0.301·11-s + 0.766·12-s − 1.33·14-s + 0.183·15-s − 0.565·16-s − 1.62·17-s − 0.508·18-s − 0.0556·19-s + 0.421·20-s + 0.504·21-s − 0.459·22-s + 1.98·23-s − 0.288·24-s − 0.899·25-s + 0.192·27-s + 1.15·28-s − 0.548·29-s − 0.279·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5577 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5577 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5577\)    =    \(3 \cdot 11 \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(44.5325\)
Root analytic conductor: \(6.67327\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{5577} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 5577,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.297400235\)
\(L(\frac12)\) \(\approx\) \(1.297400235\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - T \)
11 \( 1 - T \)
13 \( 1 \)
good2 \( 1 + 2.15T + 2T^{2} \)
5 \( 1 - 0.710T + 5T^{2} \)
7 \( 1 - 2.30T + 7T^{2} \)
17 \( 1 + 6.68T + 17T^{2} \)
19 \( 1 + 0.242T + 19T^{2} \)
23 \( 1 - 9.53T + 23T^{2} \)
29 \( 1 + 2.95T + 29T^{2} \)
31 \( 1 + 4.02T + 31T^{2} \)
37 \( 1 - 3.42T + 37T^{2} \)
41 \( 1 - 9.46T + 41T^{2} \)
43 \( 1 + 11.8T + 43T^{2} \)
47 \( 1 + 12.9T + 47T^{2} \)
53 \( 1 - 6.78T + 53T^{2} \)
59 \( 1 - 7.81T + 59T^{2} \)
61 \( 1 - 0.910T + 61T^{2} \)
67 \( 1 - 9.32T + 67T^{2} \)
71 \( 1 - 12.9T + 71T^{2} \)
73 \( 1 - 8.51T + 73T^{2} \)
79 \( 1 - 1.82T + 79T^{2} \)
83 \( 1 - 4.64T + 83T^{2} \)
89 \( 1 - 14.7T + 89T^{2} \)
97 \( 1 - 1.86T + 97T^{2} \)
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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

−8.295175286956455461468369889534, −7.70165875723914614428931530156, −6.93763799678841308376606101646, −6.47024203533367786399939931771, −5.18309488371388830089767048185, −4.53813444541758157468763651360, −3.48879313026094751394353694059, −2.24030896379559241169755955226, −1.85454842283752599940188115452, −0.75622406698047440514594154983, 0.75622406698047440514594154983, 1.85454842283752599940188115452, 2.24030896379559241169755955226, 3.48879313026094751394353694059, 4.53813444541758157468763651360, 5.18309488371388830089767048185, 6.47024203533367786399939931771, 6.93763799678841308376606101646, 7.70165875723914614428931530156, 8.295175286956455461468369889534

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