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Properties

Label 2-552-24.11-c1-0-19
Degree $2$
Conductor $552$
Sign $-0.930 - 0.365i$
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  + (1.28 + 0.595i)2-s + (−1.03 + 1.38i)3-s + (1.29 + 1.52i)4-s − 1.31·5-s + (−2.15 + 1.16i)6-s + 1.36i·7-s + (0.744 + 2.72i)8-s + (−0.855 − 2.87i)9-s + (−1.69 − 0.786i)10-s + 1.33i·11-s + (−3.45 + 0.208i)12-s + 4.58i·13-s + (−0.813 + 1.75i)14-s + (1.36 − 1.83i)15-s + (−0.670 + 3.94i)16-s − 4.76i·17-s + ⋯
L(s)  = 1  + (0.906 + 0.421i)2-s + (−0.597 + 0.801i)3-s + (0.645 + 0.764i)4-s − 0.590·5-s + (−0.879 + 0.475i)6-s + 0.516i·7-s + (0.263 + 0.964i)8-s + (−0.285 − 0.958i)9-s + (−0.535 − 0.248i)10-s + 0.403i·11-s + (−0.998 + 0.0602i)12-s + 1.27i·13-s + (−0.217 + 0.468i)14-s + (0.352 − 0.472i)15-s + (−0.167 + 0.985i)16-s − 1.15i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.930 - 0.365i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.930 - 0.365i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(552\)    =    \(2^{3} \cdot 3 \cdot 23\)
Sign: $-0.930 - 0.365i$
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: \((2,\ 552,\ (\ :1/2),\ -0.930 - 0.365i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.285855 + 1.50868i\)
\(L(\frac12)\) \(\approx\) \(0.285855 + 1.50868i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.28 - 0.595i)T \)
3 \( 1 + (1.03 - 1.38i)T \)
23 \( 1 - T \)
good5 \( 1 + 1.31T + 5T^{2} \)
7 \( 1 - 1.36iT - 7T^{2} \)
11 \( 1 - 1.33iT - 11T^{2} \)
13 \( 1 - 4.58iT - 13T^{2} \)
17 \( 1 + 4.76iT - 17T^{2} \)
19 \( 1 + 6.45T + 19T^{2} \)
29 \( 1 - 4.15T + 29T^{2} \)
31 \( 1 - 8.95iT - 31T^{2} \)
37 \( 1 + 2.96iT - 37T^{2} \)
41 \( 1 + 6.40iT - 41T^{2} \)
43 \( 1 - 12.4T + 43T^{2} \)
47 \( 1 - 4.02T + 47T^{2} \)
53 \( 1 + 9.40T + 53T^{2} \)
59 \( 1 - 4.77iT - 59T^{2} \)
61 \( 1 - 13.3iT - 61T^{2} \)
67 \( 1 - 7.21T + 67T^{2} \)
71 \( 1 - 14.4T + 71T^{2} \)
73 \( 1 - 3.85T + 73T^{2} \)
79 \( 1 + 0.671iT - 79T^{2} \)
83 \( 1 - 6.46iT - 83T^{2} \)
89 \( 1 - 4.72iT - 89T^{2} \)
97 \( 1 + 5.63T + 97T^{2} \)
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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

−11.29811383842747160133237397702, −10.63356231621322048131570067132, −9.288688724055689937076291177490, −8.589334169960985537595369731341, −7.23311828733120051551279908024, −6.52222171735018080928216186201, −5.47705658431648624351803467847, −4.53295626275478699988117619168, −3.92976118658744037343185627219, −2.49798079926615373693612821671, 0.69798356836553942086551119535, 2.28369578067786481145936470918, 3.67500183241649056755461167164, 4.65771141947300993410564616609, 5.89857301328092901527886453705, 6.40462607748608153446577771772, 7.64826396253906484550227396270, 8.226624413244876076203744234869, 9.973725187803471003735771930738, 10.93328395244259074111389657171

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