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Properties

Label 2-552-184.91-c1-0-31
Degree $2$
Conductor $552$
Sign $-0.999 - 0.0131i$
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 − 2.03·5-s + (1.38 + 0.285i)6-s + 1.21·7-s + (−2.31 − 1.62i)8-s + 9-s + (2.81 + 0.580i)10-s + 1.88i·11-s + (−1.83 − 0.791i)12-s − 3.19i·13-s + (−1.68 − 0.348i)14-s + 2.03·15-s + (2.74 + 2.90i)16-s + 5.91i·17-s + ⋯
L(s)  = 1  + (−0.979 − 0.201i)2-s − 0.577·3-s + (0.918 + 0.395i)4-s − 0.909·5-s + (0.565 + 0.116i)6-s + 0.461·7-s + (−0.819 − 0.572i)8-s + 0.333·9-s + (0.890 + 0.183i)10-s + 0.569i·11-s + (−0.530 − 0.228i)12-s − 0.884i·13-s + (−0.451 − 0.0931i)14-s + 0.524·15-s + (0.687 + 0.726i)16-s + 1.43i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0131i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0131i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(552\)    =    \(2^{3} \cdot 3 \cdot 23\)
Sign: $-0.999 - 0.0131i$
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: \((2,\ 552,\ (\ :1/2),\ -0.999 - 0.0131i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.000288292 + 0.0439175i\)
\(L(\frac12)\) \(\approx\) \(0.000288292 + 0.0439175i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.96 - 2.69i)T \)
good5 \( 1 + 2.03T + 5T^{2} \)
7 \( 1 - 1.21T + 7T^{2} \)
11 \( 1 - 1.88iT - 11T^{2} \)
13 \( 1 + 3.19iT - 13T^{2} \)
17 \( 1 - 5.91iT - 17T^{2} \)
19 \( 1 + 6.08iT - 19T^{2} \)
29 \( 1 + 9.30iT - 29T^{2} \)
31 \( 1 - 3.55iT - 31T^{2} \)
37 \( 1 + 11.1T + 37T^{2} \)
41 \( 1 + 2.52T + 41T^{2} \)
43 \( 1 - 11.6iT - 43T^{2} \)
47 \( 1 + 11.2iT - 47T^{2} \)
53 \( 1 + 11.0T + 53T^{2} \)
59 \( 1 - 0.657T + 59T^{2} \)
61 \( 1 + 10.3T + 61T^{2} \)
67 \( 1 - 2.45iT - 67T^{2} \)
71 \( 1 + 4.84iT - 71T^{2} \)
73 \( 1 + 7.87T + 73T^{2} \)
79 \( 1 - 4.23T + 79T^{2} \)
83 \( 1 - 7.58iT - 83T^{2} \)
89 \( 1 + 3.85iT - 89T^{2} \)
97 \( 1 + 3.03iT - 97T^{2} \)
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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

−10.42243010895442015847860373573, −9.611710868435403265376899977832, −8.340930814599248146098661719673, −7.88768896070432569498158458824, −6.94740695163550803952751381837, −5.92328414404867367615334189197, −4.57460453243862938937868101250, −3.39776665634709373149621777794, −1.76459604793647432495545802322, −0.03702215470231202576641258872, 1.67443756596004994875554732783, 3.44382134768630560158368625965, 4.81700335978149402298653480108, 5.91956535353300518144554760083, 6.94690306717380900376075542299, 7.68953313948971036507454467884, 8.514033923729499835534724172591, 9.403317276274144438330261307952, 10.45275158217875189071243929380, 11.13610105400015078203053893193

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