Properties

Label 2-5577-1.1-c1-0-85
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  + 1.42·2-s + 3-s + 0.0397·4-s + 0.0606·5-s + 1.42·6-s − 1.70·7-s − 2.79·8-s + 9-s + 0.0866·10-s + 11-s + 0.0397·12-s − 2.43·14-s + 0.0606·15-s − 4.07·16-s + 3.75·17-s + 1.42·18-s + 2.02·19-s + 0.00241·20-s − 1.70·21-s + 1.42·22-s + 0.704·23-s − 2.79·24-s − 4.99·25-s + 27-s − 0.0678·28-s − 0.0346·29-s + 0.0866·30-s + ⋯
L(s)  = 1  + 1.00·2-s + 0.577·3-s + 0.0198·4-s + 0.0271·5-s + 0.583·6-s − 0.644·7-s − 0.989·8-s + 0.333·9-s + 0.0273·10-s + 0.301·11-s + 0.0114·12-s − 0.651·14-s + 0.0156·15-s − 1.01·16-s + 0.910·17-s + 0.336·18-s + 0.464·19-s + 0.000538·20-s − 0.372·21-s + 0.304·22-s + 0.146·23-s − 0.571·24-s − 0.999·25-s + 0.192·27-s − 0.0128·28-s − 0.00644·29-s + 0.0158·30-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

\(L(1)\) \(\approx\) \(3.380073821\)
\(L(\frac12)\) \(\approx\) \(3.380073821\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.42T + 2T^{2} \)
5 \( 1 - 0.0606T + 5T^{2} \)
7 \( 1 + 1.70T + 7T^{2} \)
17 \( 1 - 3.75T + 17T^{2} \)
19 \( 1 - 2.02T + 19T^{2} \)
23 \( 1 - 0.704T + 23T^{2} \)
29 \( 1 + 0.0346T + 29T^{2} \)
31 \( 1 - 1.85T + 31T^{2} \)
37 \( 1 - 8.82T + 37T^{2} \)
41 \( 1 - 3.32T + 41T^{2} \)
43 \( 1 - 5.29T + 43T^{2} \)
47 \( 1 - 6.04T + 47T^{2} \)
53 \( 1 + 10.6T + 53T^{2} \)
59 \( 1 - 3.34T + 59T^{2} \)
61 \( 1 - 3.26T + 61T^{2} \)
67 \( 1 + 10.9T + 67T^{2} \)
71 \( 1 - 1.96T + 71T^{2} \)
73 \( 1 - 15.3T + 73T^{2} \)
79 \( 1 - 10.5T + 79T^{2} \)
83 \( 1 - 4.19T + 83T^{2} \)
89 \( 1 - 14.3T + 89T^{2} \)
97 \( 1 - 5.86T + 97T^{2} \)
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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

−7.998575136578154691725739516041, −7.49736585445797705893931398868, −6.41650999969050208327054292929, −6.00730256393136787087951076309, −5.16675973148142742071440830164, −4.36625470514734940075349571097, −3.65118402700268653646330611729, −3.11162645489294368959601000460, −2.24981988016235072659071400835, −0.819726463289637517392794895211, 0.819726463289637517392794895211, 2.24981988016235072659071400835, 3.11162645489294368959601000460, 3.65118402700268653646330611729, 4.36625470514734940075349571097, 5.16675973148142742071440830164, 6.00730256393136787087951076309, 6.41650999969050208327054292929, 7.49736585445797705893931398868, 7.998575136578154691725739516041

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