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Properties

Label 2-552-69.68-c3-0-11
Degree $2$
Conductor $552$
Sign $-0.998 + 0.0512i$
Analytic cond. $32.5690$
Root an. cond. $5.70693$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−5.11 − 0.890i)3-s − 20.0·5-s + 33.3i·7-s + (25.4 + 9.11i)9-s + 57.9·11-s − 50.6·13-s + (102. + 17.8i)15-s + 59.5·17-s + 71.6i·19-s + (29.6 − 170. i)21-s + (24.4 + 107. i)23-s + 277.·25-s + (−121. − 69.3i)27-s + 34.8i·29-s − 5.52·31-s + ⋯
L(s)  = 1  + (−0.985 − 0.171i)3-s − 1.79·5-s + 1.79i·7-s + (0.941 + 0.337i)9-s + 1.58·11-s − 1.08·13-s + (1.76 + 0.307i)15-s + 0.850·17-s + 0.864i·19-s + (0.308 − 1.77i)21-s + (0.221 + 0.975i)23-s + 2.21·25-s + (−0.869 − 0.493i)27-s + 0.223i·29-s − 0.0320·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0512i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.998 + 0.0512i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(552\)    =    \(2^{3} \cdot 3 \cdot 23\)
Sign: $-0.998 + 0.0512i$
Analytic conductor: \(32.5690\)
Root analytic conductor: \(5.70693\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{552} (137, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 552,\ (\ :3/2),\ -0.998 + 0.0512i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.5129379683\)
\(L(\frac12)\) \(\approx\) \(0.5129379683\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (5.11 + 0.890i)T \)
23 \( 1 + (-24.4 - 107. i)T \)
good5 \( 1 + 20.0T + 125T^{2} \)
7 \( 1 - 33.3iT - 343T^{2} \)
11 \( 1 - 57.9T + 1.33e3T^{2} \)
13 \( 1 + 50.6T + 2.19e3T^{2} \)
17 \( 1 - 59.5T + 4.91e3T^{2} \)
19 \( 1 - 71.6iT - 6.85e3T^{2} \)
29 \( 1 - 34.8iT - 2.43e4T^{2} \)
31 \( 1 + 5.52T + 2.97e4T^{2} \)
37 \( 1 - 144. iT - 5.06e4T^{2} \)
41 \( 1 - 430. iT - 6.89e4T^{2} \)
43 \( 1 + 264. iT - 7.95e4T^{2} \)
47 \( 1 - 215. iT - 1.03e5T^{2} \)
53 \( 1 + 62.8T + 1.48e5T^{2} \)
59 \( 1 - 151. iT - 2.05e5T^{2} \)
61 \( 1 - 549. iT - 2.26e5T^{2} \)
67 \( 1 + 526. iT - 3.00e5T^{2} \)
71 \( 1 + 221. iT - 3.57e5T^{2} \)
73 \( 1 + 803.T + 3.89e5T^{2} \)
79 \( 1 - 973. iT - 4.93e5T^{2} \)
83 \( 1 + 244.T + 5.71e5T^{2} \)
89 \( 1 - 966.T + 7.04e5T^{2} \)
97 \( 1 + 1.06e3iT - 9.12e5T^{2} \)
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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

−11.33828054955512193776921139607, −9.933756098782128196209925643652, −9.035147998243369883350880123721, −8.035295650861598574461997543142, −7.25216904569540098092750809335, −6.23534922984393690768221918616, −5.27607164199082106951486941738, −4.30380896503168314812774243636, −3.17705648285723660113772373344, −1.39455246320183761169231041228, 0.25584548811376933151732051498, 0.942823668266497615571461610024, 3.58563850355072197774263377105, 4.18597591822480248412006699240, 4.85673383297585718854590655109, 6.63537574018702468085047800490, 7.15389355586518987988906365931, 7.77206347930962490347560410910, 9.144172004903099739975017677558, 10.23633519540628372458909722994

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