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Properties

Label 2-552-23.18-c1-0-9
Degree $2$
Conductor $552$
Sign $-0.146 + 0.989i$
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  + (0.654 − 0.755i)3-s + (0.373 − 2.59i)5-s + (−0.503 − 1.10i)7-s + (−0.142 − 0.989i)9-s + (4.11 − 1.20i)11-s + (−0.475 + 1.04i)13-s + (−1.71 − 1.98i)15-s + (−5.64 + 3.62i)17-s + (−3.45 − 2.21i)19-s + (−1.16 − 0.341i)21-s + (1.47 − 4.56i)23-s + (−1.81 − 0.533i)25-s + (−0.841 − 0.540i)27-s + (6.89 − 4.43i)29-s + (−4.60 − 5.31i)31-s + ⋯
L(s)  = 1  + (0.378 − 0.436i)3-s + (0.167 − 1.16i)5-s + (−0.190 − 0.416i)7-s + (−0.0474 − 0.329i)9-s + (1.23 − 0.364i)11-s + (−0.132 + 0.289i)13-s + (−0.443 − 0.512i)15-s + (−1.36 + 0.879i)17-s + (−0.791 − 0.508i)19-s + (−0.253 − 0.0745i)21-s + (0.307 − 0.951i)23-s + (−0.363 − 0.106i)25-s + (−0.161 − 0.104i)27-s + (1.28 − 0.823i)29-s + (−0.827 − 0.955i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.146 + 0.989i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.146 + 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(552\)    =    \(2^{3} \cdot 3 \cdot 23\)
Sign: $-0.146 + 0.989i$
Analytic conductor: \(4.40774\)
Root analytic conductor: \(2.09946\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{552} (409, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 552,\ (\ :1/2),\ -0.146 + 0.989i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.03804 - 1.20322i\)
\(L(\frac12)\) \(\approx\) \(1.03804 - 1.20322i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.654 + 0.755i)T \)
23 \( 1 + (-1.47 + 4.56i)T \)
good5 \( 1 + (-0.373 + 2.59i)T + (-4.79 - 1.40i)T^{2} \)
7 \( 1 + (0.503 + 1.10i)T + (-4.58 + 5.29i)T^{2} \)
11 \( 1 + (-4.11 + 1.20i)T + (9.25 - 5.94i)T^{2} \)
13 \( 1 + (0.475 - 1.04i)T + (-8.51 - 9.82i)T^{2} \)
17 \( 1 + (5.64 - 3.62i)T + (7.06 - 15.4i)T^{2} \)
19 \( 1 + (3.45 + 2.21i)T + (7.89 + 17.2i)T^{2} \)
29 \( 1 + (-6.89 + 4.43i)T + (12.0 - 26.3i)T^{2} \)
31 \( 1 + (4.60 + 5.31i)T + (-4.41 + 30.6i)T^{2} \)
37 \( 1 + (-0.293 - 2.03i)T + (-35.5 + 10.4i)T^{2} \)
41 \( 1 + (1.36 - 9.46i)T + (-39.3 - 11.5i)T^{2} \)
43 \( 1 + (1.90 - 2.20i)T + (-6.11 - 42.5i)T^{2} \)
47 \( 1 - 7.43T + 47T^{2} \)
53 \( 1 + (-5.01 - 10.9i)T + (-34.7 + 40.0i)T^{2} \)
59 \( 1 + (-3.56 + 7.80i)T + (-38.6 - 44.5i)T^{2} \)
61 \( 1 + (2.15 + 2.48i)T + (-8.68 + 60.3i)T^{2} \)
67 \( 1 + (-1.61 - 0.474i)T + (56.3 + 36.2i)T^{2} \)
71 \( 1 + (-3.54 - 1.04i)T + (59.7 + 38.3i)T^{2} \)
73 \( 1 + (0.819 + 0.526i)T + (30.3 + 66.4i)T^{2} \)
79 \( 1 + (-3.11 + 6.81i)T + (-51.7 - 59.7i)T^{2} \)
83 \( 1 + (-1.05 - 7.31i)T + (-79.6 + 23.3i)T^{2} \)
89 \( 1 + (-4.11 + 4.74i)T + (-12.6 - 88.0i)T^{2} \)
97 \( 1 + (2.20 - 15.3i)T + (-93.0 - 27.3i)T^{2} \)
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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

−10.55122268752669545065964476911, −9.318730132537211503752664702024, −8.823838844051142298131811626911, −8.121391493898509908740431854979, −6.72676989747959989222539925687, −6.23653361006097323214148768974, −4.64376828390830071926092701999, −3.99240068634714498262259339567, −2.26976400476022250490008861171, −0.914533218328591890090341954086, 2.12196579475667546526905241730, 3.18564246933514422642645866092, 4.21324524620794082696485806965, 5.49958205602374130868213331635, 6.75090138322270188870615151713, 7.12256800158982286407645001737, 8.693143526760611427840998596212, 9.176867298729945464202512118171, 10.23603440185692680726441707751, 10.84746425699246852948911543313

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