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Properties

Label 2-552-552.413-c1-0-44
Degree $2$
Conductor $552$
Sign $0.986 - 0.166i$
Analytic cond. $4.40774$
Root an. cond. $2.09946$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.261 + 1.38i)2-s + (−1.60 − 0.661i)3-s + (−1.86 + 0.728i)4-s + (0.499 − 2.39i)6-s + (−1.49 − 2.39i)8-s + (2.12 + 2.11i)9-s + (3.46 + 0.0666i)12-s − 6.88i·13-s + (2.93 − 2.71i)16-s + (−2.38 + 3.50i)18-s + 4.79i·23-s + (0.814 + 4.83i)24-s + 5·25-s + (9.56 − 1.80i)26-s + (−2.00 − 4.79i)27-s + ⋯
L(s)  = 1  + (0.185 + 0.982i)2-s + (−0.924 − 0.381i)3-s + (−0.931 + 0.364i)4-s + (0.204 − 0.978i)6-s + (−0.530 − 0.847i)8-s + (0.708 + 0.705i)9-s + (0.999 + 0.0192i)12-s − 1.90i·13-s + (0.734 − 0.678i)16-s + (−0.562 + 0.826i)18-s + 0.999i·23-s + (0.166 + 0.986i)24-s + 25-s + (1.87 − 0.353i)26-s + (−0.384 − 0.922i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.986 - 0.166i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.986 - 0.166i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(552\)    =    \(2^{3} \cdot 3 \cdot 23\)
Sign: $0.986 - 0.166i$
Analytic conductor: \(4.40774\)
Root analytic conductor: \(2.09946\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{552} (413, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 552,\ (\ :1/2),\ 0.986 - 0.166i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.982601 + 0.0822834i\)
\(L(\frac12)\) \(\approx\) \(0.982601 + 0.0822834i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.261 - 1.38i)T \)
3 \( 1 + (1.60 + 0.661i)T \)
23 \( 1 - 4.79iT \)
good5 \( 1 - 5T^{2} \)
7 \( 1 - 7T^{2} \)
11 \( 1 - 11T^{2} \)
13 \( 1 + 6.88iT - 13T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 + 19T^{2} \)
29 \( 1 - 10.6T + 29T^{2} \)
31 \( 1 - 5.29T + 31T^{2} \)
37 \( 1 + 37T^{2} \)
41 \( 1 + 9.52iT - 41T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 + 7.14iT - 47T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + 12T + 59T^{2} \)
61 \( 1 + 61T^{2} \)
67 \( 1 + 67T^{2} \)
71 \( 1 + 15.0iT - 71T^{2} \)
73 \( 1 + 17.0T + 73T^{2} \)
79 \( 1 - 79T^{2} \)
83 \( 1 - 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 - 97T^{2} \)
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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

−10.59957590734457601750617939876, −10.11896574836594563745389367548, −8.749287075450946503741592709495, −7.88271166777670968372280774701, −7.13885439214209422763614009948, −6.16199893898104552021643528646, −5.42172844991354223041747103654, −4.62079357500161832176779787366, −3.12123940796044976398457012847, −0.74690058599252993485919153500, 1.21276530993520022855550694278, 2.82479845461763649290073598976, 4.45076735910433631161509513338, 4.59225767766055880841198892869, 6.09633723165888743800418684524, 6.81157018907646757813517825744, 8.485090858057933182548171978380, 9.289692004575242558701582150800, 10.12866641713021014147499602722, 10.82092113028604318961163152049

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