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Properties

Label 2-552-8.5-c1-0-33
Degree $2$
Conductor $552$
Sign $0.823 + 0.567i$
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  + (1.34 − 0.447i)2-s + i·3-s + (1.59 − 1.20i)4-s − 1.69i·5-s + (0.447 + 1.34i)6-s + 2.41·7-s + (1.60 − 2.32i)8-s − 9-s + (−0.761 − 2.27i)10-s + 0.133i·11-s + (1.20 + 1.59i)12-s − 0.529i·13-s + (3.23 − 1.08i)14-s + 1.69·15-s + (1.11 − 3.84i)16-s − 3.72·17-s + ⋯
L(s)  = 1  + (0.948 − 0.316i)2-s + 0.577i·3-s + (0.799 − 0.600i)4-s − 0.760i·5-s + (0.182 + 0.547i)6-s + 0.912·7-s + (0.567 − 0.823i)8-s − 0.333·9-s + (−0.240 − 0.720i)10-s + 0.0401i·11-s + (0.346 + 0.461i)12-s − 0.146i·13-s + (0.865 − 0.289i)14-s + 0.438·15-s + (0.278 − 0.960i)16-s − 0.903·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.823 + 0.567i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.823 + 0.567i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(552\)    =    \(2^{3} \cdot 3 \cdot 23\)
Sign: $0.823 + 0.567i$
Analytic conductor: \(4.40774\)
Root analytic conductor: \(2.09946\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{552} (277, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 552,\ (\ :1/2),\ 0.823 + 0.567i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.62006 - 0.816209i\)
\(L(\frac12)\) \(\approx\) \(2.62006 - 0.816209i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.34 + 0.447i)T \)
3 \( 1 - iT \)
23 \( 1 - T \)
good5 \( 1 + 1.69iT - 5T^{2} \)
7 \( 1 - 2.41T + 7T^{2} \)
11 \( 1 - 0.133iT - 11T^{2} \)
13 \( 1 + 0.529iT - 13T^{2} \)
17 \( 1 + 3.72T + 17T^{2} \)
19 \( 1 - 1.26iT - 19T^{2} \)
29 \( 1 - 2.58iT - 29T^{2} \)
31 \( 1 - 4.37T + 31T^{2} \)
37 \( 1 - 3.36iT - 37T^{2} \)
41 \( 1 + 5.68T + 41T^{2} \)
43 \( 1 - 2.62iT - 43T^{2} \)
47 \( 1 + 0.552T + 47T^{2} \)
53 \( 1 - 10.6iT - 53T^{2} \)
59 \( 1 + 3.59iT - 59T^{2} \)
61 \( 1 - 6.75iT - 61T^{2} \)
67 \( 1 - 0.405iT - 67T^{2} \)
71 \( 1 + 5.62T + 71T^{2} \)
73 \( 1 + 8.66T + 73T^{2} \)
79 \( 1 + 2.77T + 79T^{2} \)
83 \( 1 - 1.80iT - 83T^{2} \)
89 \( 1 + 14.1T + 89T^{2} \)
97 \( 1 - 5.11T + 97T^{2} \)
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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

−10.87555420438530620297995931330, −10.06688577652505306711356044890, −8.978769703508626920294101085879, −8.129400470169362061524314756809, −6.88690804330639714343386267312, −5.71690827076459059227866364019, −4.81042031123102470920792414163, −4.32265398934825406336614454623, −2.92185226955343166934717621606, −1.45756527664589337612223891607, 1.93535474552577589910394729944, 2.99076484094765473824418284237, 4.32718839269079236140179512390, 5.29116071034074319520293946189, 6.44697531642386465719510124957, 7.02132475440379461514499997676, 7.945682096509684529061250450700, 8.761443075482908886190557926936, 10.34483765008283238607117306462, 11.25864403495544905642933358074

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