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Properties

Label 2-552-184.91-c1-0-44
Degree $2$
Conductor $552$
Sign $-0.0328 + 0.999i$
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  + (1.38 − 0.285i)2-s − 3-s + (1.83 − 0.791i)4-s − 2.80·5-s + (−1.38 + 0.285i)6-s + 0.415·7-s + (2.31 − 1.62i)8-s + 9-s + (−3.88 + 0.801i)10-s − 5.54i·11-s + (−1.83 + 0.791i)12-s − 3.19i·13-s + (0.575 − 0.118i)14-s + 2.80·15-s + (2.74 − 2.90i)16-s − 2.01i·17-s + ⋯
L(s)  = 1  + (0.979 − 0.201i)2-s − 0.577·3-s + (0.918 − 0.395i)4-s − 1.25·5-s + (−0.565 + 0.116i)6-s + 0.156·7-s + (0.819 − 0.572i)8-s + 0.333·9-s + (−1.22 + 0.253i)10-s − 1.67i·11-s + (−0.530 + 0.228i)12-s − 0.884i·13-s + (0.153 − 0.0316i)14-s + 0.724·15-s + (0.687 − 0.726i)16-s − 0.488i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0328 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0328 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(552\)    =    \(2^{3} \cdot 3 \cdot 23\)
Sign: $-0.0328 + 0.999i$
Analytic conductor: \(4.40774\)
Root analytic conductor: \(2.09946\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{552} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 552,\ (\ :1/2),\ -0.0328 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.17445 - 1.21369i\)
\(L(\frac12)\) \(\approx\) \(1.17445 - 1.21369i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.38 + 0.285i)T \)
3 \( 1 + T \)
23 \( 1 + (-2.87 + 3.83i)T \)
good5 \( 1 + 2.80T + 5T^{2} \)
7 \( 1 - 0.415T + 7T^{2} \)
11 \( 1 + 5.54iT - 11T^{2} \)
13 \( 1 + 3.19iT - 13T^{2} \)
17 \( 1 + 2.01iT - 17T^{2} \)
19 \( 1 - 2.50iT - 19T^{2} \)
29 \( 1 + 4.69iT - 29T^{2} \)
31 \( 1 - 8.16iT - 31T^{2} \)
37 \( 1 + 3.59T + 37T^{2} \)
41 \( 1 - 2.52T + 41T^{2} \)
43 \( 1 - 1.80iT - 43T^{2} \)
47 \( 1 - 3.58iT - 47T^{2} \)
53 \( 1 - 0.270T + 53T^{2} \)
59 \( 1 + 0.657T + 59T^{2} \)
61 \( 1 - 11.8T + 61T^{2} \)
67 \( 1 - 5.41iT - 67T^{2} \)
71 \( 1 - 6.09iT - 71T^{2} \)
73 \( 1 + 2.82T + 73T^{2} \)
79 \( 1 - 15.0T + 79T^{2} \)
83 \( 1 - 11.1iT - 83T^{2} \)
89 \( 1 + 4.02iT - 89T^{2} \)
97 \( 1 + 14.5iT - 97T^{2} \)
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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

−11.01830082864843102716300970007, −10.14111450945875531567635478678, −8.522265075354532284692722667983, −7.77140736900522240575913472731, −6.72993840325169006673751336606, −5.76827603100347254810081986314, −4.90670499683607050984754034025, −3.79651785492705152873766534523, −2.98644112710939709939302066995, −0.78567174576563836237184172073, 1.95725359022427271398878371521, 3.66255584537112630928122576890, 4.45077343665817283457831468452, 5.16275729566694698061331968095, 6.56542557286578970568976215712, 7.23721257654159066169007135853, 7.891319862649012640154021541576, 9.272732342342401729358405441065, 10.48835769463863802275568636656, 11.42245705739263146168871280379

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