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Properties

Label 2-552-24.11-c1-0-21
Degree $2$
Conductor $552$
Sign $-0.142 - 0.989i$
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  + (−1.41 + 0.0336i)2-s + (0.124 + 1.72i)3-s + (1.99 − 0.0951i)4-s + 1.31·5-s + (−0.233 − 2.43i)6-s − 2.00i·7-s + (−2.82 + 0.201i)8-s + (−2.96 + 0.429i)9-s + (−1.86 + 0.0443i)10-s + 2.03i·11-s + (0.412 + 3.43i)12-s + 2.24i·13-s + (0.0675 + 2.84i)14-s + (0.163 + 2.27i)15-s + (3.98 − 0.380i)16-s + 5.84i·17-s + ⋯
L(s)  = 1  + (−0.999 + 0.0237i)2-s + (0.0717 + 0.997i)3-s + (0.998 − 0.0475i)4-s + 0.588·5-s + (−0.0954 − 0.995i)6-s − 0.759i·7-s + (−0.997 + 0.0713i)8-s + (−0.989 + 0.143i)9-s + (−0.588 + 0.0140i)10-s + 0.615i·11-s + (0.119 + 0.992i)12-s + 0.623i·13-s + (0.0180 + 0.759i)14-s + (0.0422 + 0.587i)15-s + (0.995 − 0.0950i)16-s + 1.41i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.142 - 0.989i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.142 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(552\)    =    \(2^{3} \cdot 3 \cdot 23\)
Sign: $-0.142 - 0.989i$
Analytic conductor: \(4.40774\)
Root analytic conductor: \(2.09946\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{552} (323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 552,\ (\ :1/2),\ -0.142 - 0.989i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.628884 + 0.726050i\)
\(L(\frac12)\) \(\approx\) \(0.628884 + 0.726050i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.41 - 0.0336i)T \)
3 \( 1 + (-0.124 - 1.72i)T \)
23 \( 1 - T \)
good5 \( 1 - 1.31T + 5T^{2} \)
7 \( 1 + 2.00iT - 7T^{2} \)
11 \( 1 - 2.03iT - 11T^{2} \)
13 \( 1 - 2.24iT - 13T^{2} \)
17 \( 1 - 5.84iT - 17T^{2} \)
19 \( 1 - 5.65T + 19T^{2} \)
29 \( 1 - 0.456T + 29T^{2} \)
31 \( 1 - 4.61iT - 31T^{2} \)
37 \( 1 - 10.8iT - 37T^{2} \)
41 \( 1 - 1.82iT - 41T^{2} \)
43 \( 1 - 7.89T + 43T^{2} \)
47 \( 1 - 5.66T + 47T^{2} \)
53 \( 1 + 10.1T + 53T^{2} \)
59 \( 1 + 9.04iT - 59T^{2} \)
61 \( 1 + 3.19iT - 61T^{2} \)
67 \( 1 - 0.607T + 67T^{2} \)
71 \( 1 + 7.64T + 71T^{2} \)
73 \( 1 + 13.1T + 73T^{2} \)
79 \( 1 - 9.56iT - 79T^{2} \)
83 \( 1 - 1.39iT - 83T^{2} \)
89 \( 1 - 5.14iT - 89T^{2} \)
97 \( 1 - 0.534T + 97T^{2} \)
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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

−10.69061645413862773003863443010, −9.981662656235010427819557805499, −9.527299495124978704562960820996, −8.568511641776537934693764176496, −7.63595137346425067963172028511, −6.57763151378092253647592170499, −5.60518949195840572923505420407, −4.32478433748027083119977554225, −3.11405616493608826999849019186, −1.60326573653234125929927405985, 0.78032053176158454799929953142, 2.26698849243378507900418503911, 3.05939596735995167922843322677, 5.61907800417297274675363426890, 5.91177668193259877119785449255, 7.29538537880708281426492818059, 7.73126950039723028593002218793, 9.004206099886212816584041833672, 9.277232961523597254951161615170, 10.50000132037040468334227120074

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