Properties

Label 2-5577-1.1-c1-0-197
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  + 2.73·2-s + 3-s + 5.50·4-s − 2.84·5-s + 2.73·6-s + 3.93·7-s + 9.58·8-s + 9-s − 7.78·10-s + 11-s + 5.50·12-s + 10.7·14-s − 2.84·15-s + 15.2·16-s + 3.81·17-s + 2.73·18-s − 2.94·19-s − 15.6·20-s + 3.93·21-s + 2.73·22-s − 1.89·23-s + 9.58·24-s + 3.07·25-s + 27-s + 21.6·28-s − 2.09·29-s − 7.78·30-s + ⋯
L(s)  = 1  + 1.93·2-s + 0.577·3-s + 2.75·4-s − 1.27·5-s + 1.11·6-s + 1.48·7-s + 3.38·8-s + 0.333·9-s − 2.46·10-s + 0.301·11-s + 1.58·12-s + 2.87·14-s − 0.733·15-s + 3.81·16-s + 0.926·17-s + 0.645·18-s − 0.674·19-s − 3.49·20-s + 0.857·21-s + 0.583·22-s − 0.395·23-s + 1.95·24-s + 0.614·25-s + 0.192·27-s + 4.08·28-s − 0.389·29-s − 1.42·30-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

\(L(1)\) \(\approx\) \(9.211795358\)
\(L(\frac12)\) \(\approx\) \(9.211795358\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.73T + 2T^{2} \)
5 \( 1 + 2.84T + 5T^{2} \)
7 \( 1 - 3.93T + 7T^{2} \)
17 \( 1 - 3.81T + 17T^{2} \)
19 \( 1 + 2.94T + 19T^{2} \)
23 \( 1 + 1.89T + 23T^{2} \)
29 \( 1 + 2.09T + 29T^{2} \)
31 \( 1 + 6.16T + 31T^{2} \)
37 \( 1 - 8.34T + 37T^{2} \)
41 \( 1 - 6.35T + 41T^{2} \)
43 \( 1 + 11.7T + 43T^{2} \)
47 \( 1 - 5.31T + 47T^{2} \)
53 \( 1 + 2.37T + 53T^{2} \)
59 \( 1 + 5.38T + 59T^{2} \)
61 \( 1 + 2.34T + 61T^{2} \)
67 \( 1 - 10.4T + 67T^{2} \)
71 \( 1 - 10.0T + 71T^{2} \)
73 \( 1 + 15.1T + 73T^{2} \)
79 \( 1 - 1.57T + 79T^{2} \)
83 \( 1 - 10.6T + 83T^{2} \)
89 \( 1 - 3.23T + 89T^{2} \)
97 \( 1 + 17.7T + 97T^{2} \)
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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

−7.892838682134381843111245174970, −7.45454971939032172284494616512, −6.66911378224159007786151717799, −5.70791805263340230571278081398, −5.04126758109308459294655539580, −4.24339052114726560853597812940, −3.97149821567742913286491795951, −3.16200787216044213230439990459, −2.20191019366229916208339818894, −1.36815985258767532950655311011, 1.36815985258767532950655311011, 2.20191019366229916208339818894, 3.16200787216044213230439990459, 3.97149821567742913286491795951, 4.24339052114726560853597812940, 5.04126758109308459294655539580, 5.70791805263340230571278081398, 6.66911378224159007786151717799, 7.45454971939032172284494616512, 7.892838682134381843111245174970

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