Properties

Label 2-5577-1.1-c1-0-190
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 $1$

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: \(1\)
Selberg data: \((2,\ 5577,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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

−8.044545333471564025909506275058, −7.15015501634060252107063393008, −6.64632158672489329068713394782, −6.03739762323522552777652384289, −5.29844280461201546924575155875, −3.36994325057554322143589480833, −3.00827588802802444326164293418, −2.05733419985698124414175194443, −1.32014839304408894187199495511, 0, 1.32014839304408894187199495511, 2.05733419985698124414175194443, 3.00827588802802444326164293418, 3.36994325057554322143589480833, 5.29844280461201546924575155875, 6.03739762323522552777652384289, 6.64632158672489329068713394782, 7.15015501634060252107063393008, 8.044545333471564025909506275058

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