Properties

Label 2-5577-1.1-c1-0-113
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  − 1.65·2-s + 3-s + 0.723·4-s + 4.43·5-s − 1.65·6-s − 3.18·7-s + 2.10·8-s + 9-s − 7.31·10-s + 11-s + 0.723·12-s + 5.26·14-s + 4.43·15-s − 4.92·16-s − 1.25·17-s − 1.65·18-s + 5.97·19-s + 3.20·20-s − 3.18·21-s − 1.65·22-s + 6.94·23-s + 2.10·24-s + 14.6·25-s + 27-s − 2.30·28-s + 5.43·29-s − 7.31·30-s + ⋯
L(s)  = 1  − 1.16·2-s + 0.577·3-s + 0.361·4-s + 1.98·5-s − 0.673·6-s − 1.20·7-s + 0.744·8-s + 0.333·9-s − 2.31·10-s + 0.301·11-s + 0.208·12-s + 1.40·14-s + 1.14·15-s − 1.23·16-s − 0.303·17-s − 0.389·18-s + 1.37·19-s + 0.717·20-s − 0.695·21-s − 0.351·22-s + 1.44·23-s + 0.429·24-s + 2.92·25-s + 0.192·27-s − 0.435·28-s + 1.00·29-s − 1.33·30-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 5577,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.903312805\)
\(L(\frac12)\) \(\approx\) \(1.903312805\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.65T + 2T^{2} \)
5 \( 1 - 4.43T + 5T^{2} \)
7 \( 1 + 3.18T + 7T^{2} \)
17 \( 1 + 1.25T + 17T^{2} \)
19 \( 1 - 5.97T + 19T^{2} \)
23 \( 1 - 6.94T + 23T^{2} \)
29 \( 1 - 5.43T + 29T^{2} \)
31 \( 1 - 7.54T + 31T^{2} \)
37 \( 1 - 2.52T + 37T^{2} \)
41 \( 1 + 10.7T + 41T^{2} \)
43 \( 1 - 0.983T + 43T^{2} \)
47 \( 1 + 11.7T + 47T^{2} \)
53 \( 1 - 6.73T + 53T^{2} \)
59 \( 1 + 2.73T + 59T^{2} \)
61 \( 1 + 2.73T + 61T^{2} \)
67 \( 1 + 9.44T + 67T^{2} \)
71 \( 1 - 6.30T + 71T^{2} \)
73 \( 1 - 1.95T + 73T^{2} \)
79 \( 1 + 9.68T + 79T^{2} \)
83 \( 1 + 1.47T + 83T^{2} \)
89 \( 1 + 2.64T + 89T^{2} \)
97 \( 1 - 2.66T + 97T^{2} \)
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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

−8.592201509860797587378696311245, −7.46257466304353667362349534299, −6.64591245676032661362053108769, −6.43588858043326419754875003981, −5.29636536562714305347493794326, −4.66429984621287100090779533358, −3.18467257190384813324727230509, −2.71669046473187743720664046702, −1.62809945324492102102345549169, −0.931082693821154427987442061988, 0.931082693821154427987442061988, 1.62809945324492102102345549169, 2.71669046473187743720664046702, 3.18467257190384813324727230509, 4.66429984621287100090779533358, 5.29636536562714305347493794326, 6.43588858043326419754875003981, 6.64591245676032661362053108769, 7.46257466304353667362349534299, 8.592201509860797587378696311245

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