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Properties

Label 2-552-184.91-c1-0-23
Degree $2$
Conductor $552$
Sign $0.203 - 0.979i$
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  + (0.0902 + 1.41i)2-s + 3-s + (−1.98 + 0.254i)4-s + 3.06·5-s + (0.0902 + 1.41i)6-s + 0.892·7-s + (−0.538 − 2.77i)8-s + 9-s + (0.276 + 4.32i)10-s + 3.95i·11-s + (−1.98 + 0.254i)12-s − 4.14i·13-s + (0.0804 + 1.25i)14-s + 3.06·15-s + (3.87 − 1.01i)16-s + 1.01i·17-s + ⋯
L(s)  = 1  + (0.0637 + 0.997i)2-s + 0.577·3-s + (−0.991 + 0.127i)4-s + 1.36·5-s + (0.0368 + 0.576i)6-s + 0.337·7-s + (−0.190 − 0.981i)8-s + 0.333·9-s + (0.0873 + 1.36i)10-s + 1.19i·11-s + (−0.572 + 0.0735i)12-s − 1.15i·13-s + (0.0215 + 0.336i)14-s + 0.790·15-s + (0.967 − 0.252i)16-s + 0.245i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.203 - 0.979i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.203 - 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(552\)    =    \(2^{3} \cdot 3 \cdot 23\)
Sign: $0.203 - 0.979i$
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: \((2,\ 552,\ (\ :1/2),\ 0.203 - 0.979i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.62645 + 1.32343i\)
\(L(\frac12)\) \(\approx\) \(1.62645 + 1.32343i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.0902 - 1.41i)T \)
3 \( 1 - T \)
23 \( 1 + (-4.42 - 1.85i)T \)
good5 \( 1 - 3.06T + 5T^{2} \)
7 \( 1 - 0.892T + 7T^{2} \)
11 \( 1 - 3.95iT - 11T^{2} \)
13 \( 1 + 4.14iT - 13T^{2} \)
17 \( 1 - 1.01iT - 17T^{2} \)
19 \( 1 + 0.195iT - 19T^{2} \)
29 \( 1 + 3.05iT - 29T^{2} \)
31 \( 1 - 2.58iT - 31T^{2} \)
37 \( 1 + 6.01T + 37T^{2} \)
41 \( 1 + 10.2T + 41T^{2} \)
43 \( 1 - 10.0iT - 43T^{2} \)
47 \( 1 + 7.33iT - 47T^{2} \)
53 \( 1 - 11.2T + 53T^{2} \)
59 \( 1 + 4.35T + 59T^{2} \)
61 \( 1 + 7.63T + 61T^{2} \)
67 \( 1 + 3.60iT - 67T^{2} \)
71 \( 1 + 10.5iT - 71T^{2} \)
73 \( 1 - 3.24T + 73T^{2} \)
79 \( 1 + 1.85T + 79T^{2} \)
83 \( 1 + 3.79iT - 83T^{2} \)
89 \( 1 + 10.7iT - 89T^{2} \)
97 \( 1 + 2.49iT - 97T^{2} \)
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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

−10.46476022807704391311184479320, −9.905959611518083680762618108057, −9.153994307460595129058873239225, −8.264729866181746814962765293796, −7.35996102126576014121822841513, −6.48558748388224637320850281743, −5.42678188273139621379409994376, −4.73390939105692270420537774616, −3.20594905499677597953511069568, −1.70007733865286437157426163092, 1.43463409311113971576008311823, 2.43686752537118054810794454323, 3.54115726169751187427700063075, 4.84573891348383215382379475415, 5.75449514711243336161897431374, 6.91837749518947206680176805142, 8.487057624320594539938336007828, 8.950763691553158968669272633490, 9.732399912971247052741649130155, 10.57316238551159945276638185044

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