Properties

Label 2-5577-1.1-c1-0-70
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  − 0.584·2-s + 3-s − 1.65·4-s − 1.95·5-s − 0.584·6-s + 1.51·7-s + 2.13·8-s + 9-s + 1.14·10-s + 11-s − 1.65·12-s − 0.883·14-s − 1.95·15-s + 2.06·16-s + 3.44·17-s − 0.584·18-s + 5.09·19-s + 3.23·20-s + 1.51·21-s − 0.584·22-s − 0.701·23-s + 2.13·24-s − 1.18·25-s + 27-s − 2.50·28-s + 5.04·29-s + 1.14·30-s + ⋯
L(s)  = 1  − 0.413·2-s + 0.577·3-s − 0.829·4-s − 0.873·5-s − 0.238·6-s + 0.571·7-s + 0.756·8-s + 0.333·9-s + 0.361·10-s + 0.301·11-s − 0.478·12-s − 0.236·14-s − 0.504·15-s + 0.516·16-s + 0.834·17-s − 0.137·18-s + 1.16·19-s + 0.724·20-s + 0.329·21-s − 0.124·22-s − 0.146·23-s + 0.436·24-s − 0.236·25-s + 0.192·27-s − 0.473·28-s + 0.937·29-s + 0.208·30-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

\(L(1)\) \(\approx\) \(1.546012810\)
\(L(\frac12)\) \(\approx\) \(1.546012810\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.584T + 2T^{2} \)
5 \( 1 + 1.95T + 5T^{2} \)
7 \( 1 - 1.51T + 7T^{2} \)
17 \( 1 - 3.44T + 17T^{2} \)
19 \( 1 - 5.09T + 19T^{2} \)
23 \( 1 + 0.701T + 23T^{2} \)
29 \( 1 - 5.04T + 29T^{2} \)
31 \( 1 - 7.26T + 31T^{2} \)
37 \( 1 + 2.08T + 37T^{2} \)
41 \( 1 - 1.03T + 41T^{2} \)
43 \( 1 + 5.48T + 43T^{2} \)
47 \( 1 - 3.75T + 47T^{2} \)
53 \( 1 + 0.502T + 53T^{2} \)
59 \( 1 + 2.65T + 59T^{2} \)
61 \( 1 + 5.38T + 61T^{2} \)
67 \( 1 + 8.47T + 67T^{2} \)
71 \( 1 + 2.31T + 71T^{2} \)
73 \( 1 - 2.21T + 73T^{2} \)
79 \( 1 + 5.54T + 79T^{2} \)
83 \( 1 - 14.5T + 83T^{2} \)
89 \( 1 + 1.80T + 89T^{2} \)
97 \( 1 - 14.5T + 97T^{2} \)
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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

−7.993533607860616646353911846472, −7.85589252039131430546566230423, −7.06450031960879121405287741528, −5.97607718084513049149257360088, −4.96942492536879530660471230012, −4.49854173974848639362389679356, −3.64933768369989421152563703771, −3.02585224490340776570661532012, −1.60769981803563766395743158731, −0.74649164071958294395262759525, 0.74649164071958294395262759525, 1.60769981803563766395743158731, 3.02585224490340776570661532012, 3.64933768369989421152563703771, 4.49854173974848639362389679356, 4.96942492536879530660471230012, 5.97607718084513049149257360088, 7.06450031960879121405287741528, 7.85589252039131430546566230423, 7.993533607860616646353911846472

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