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Properties

Label 2-552-23.16-c1-0-0
Degree $2$
Conductor $552$
Sign $-0.0266 - 0.999i$
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  + (−0.841 + 0.540i)3-s + (−1.14 − 2.51i)5-s + (−2.52 + 0.740i)7-s + (0.415 − 0.909i)9-s + (3.07 + 3.54i)11-s + (−1.45 − 0.427i)13-s + (2.32 + 1.49i)15-s + (−0.675 + 4.69i)17-s + (0.942 + 6.55i)19-s + (1.72 − 1.98i)21-s + (4.79 − 0.184i)23-s + (−1.73 + 2.00i)25-s + (0.142 + 0.989i)27-s + (−1.28 + 8.94i)29-s + (−3.58 − 2.30i)31-s + ⋯
L(s)  = 1  + (−0.485 + 0.312i)3-s + (−0.513 − 1.12i)5-s + (−0.953 + 0.279i)7-s + (0.138 − 0.303i)9-s + (0.925 + 1.06i)11-s + (−0.403 − 0.118i)13-s + (0.600 + 0.386i)15-s + (−0.163 + 1.13i)17-s + (0.216 + 1.50i)19-s + (0.375 − 0.433i)21-s + (0.999 − 0.0384i)23-s + (−0.346 + 0.400i)25-s + (0.0273 + 0.190i)27-s + (−0.238 + 1.66i)29-s + (−0.643 − 0.413i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0266 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0266 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(552\)    =    \(2^{3} \cdot 3 \cdot 23\)
Sign: $-0.0266 - 0.999i$
Analytic conductor: \(4.40774\)
Root analytic conductor: \(2.09946\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{552} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 552,\ (\ :1/2),\ -0.0266 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.506727 + 0.520413i\)
\(L(\frac12)\) \(\approx\) \(0.506727 + 0.520413i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.841 - 0.540i)T \)
23 \( 1 + (-4.79 + 0.184i)T \)
good5 \( 1 + (1.14 + 2.51i)T + (-3.27 + 3.77i)T^{2} \)
7 \( 1 + (2.52 - 0.740i)T + (5.88 - 3.78i)T^{2} \)
11 \( 1 + (-3.07 - 3.54i)T + (-1.56 + 10.8i)T^{2} \)
13 \( 1 + (1.45 + 0.427i)T + (10.9 + 7.02i)T^{2} \)
17 \( 1 + (0.675 - 4.69i)T + (-16.3 - 4.78i)T^{2} \)
19 \( 1 + (-0.942 - 6.55i)T + (-18.2 + 5.35i)T^{2} \)
29 \( 1 + (1.28 - 8.94i)T + (-27.8 - 8.17i)T^{2} \)
31 \( 1 + (3.58 + 2.30i)T + (12.8 + 28.1i)T^{2} \)
37 \( 1 + (-2.38 + 5.21i)T + (-24.2 - 27.9i)T^{2} \)
41 \( 1 + (-0.138 - 0.302i)T + (-26.8 + 30.9i)T^{2} \)
43 \( 1 + (8.28 - 5.32i)T + (17.8 - 39.1i)T^{2} \)
47 \( 1 - 3.20T + 47T^{2} \)
53 \( 1 + (4.88 - 1.43i)T + (44.5 - 28.6i)T^{2} \)
59 \( 1 + (-4.56 - 1.34i)T + (49.6 + 31.8i)T^{2} \)
61 \( 1 + (0.723 + 0.464i)T + (25.3 + 55.4i)T^{2} \)
67 \( 1 + (-0.547 + 0.632i)T + (-9.53 - 66.3i)T^{2} \)
71 \( 1 + (10.1 - 11.7i)T + (-10.1 - 70.2i)T^{2} \)
73 \( 1 + (0.826 + 5.74i)T + (-70.0 + 20.5i)T^{2} \)
79 \( 1 + (-14.5 - 4.28i)T + (66.4 + 42.7i)T^{2} \)
83 \( 1 + (4.17 - 9.14i)T + (-54.3 - 62.7i)T^{2} \)
89 \( 1 + (-11.6 + 7.48i)T + (36.9 - 80.9i)T^{2} \)
97 \( 1 + (2.56 + 5.62i)T + (-63.5 + 73.3i)T^{2} \)
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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

−11.02513476782767098929965663354, −9.973146776654868294861565231970, −9.328601979929303321815351668253, −8.543232078176682222706887655000, −7.36914543558732780132575180623, −6.38239232846945967207978285467, −5.38022314351548127864460992336, −4.38588412159672586160602978524, −3.52554488275129706130706657340, −1.48802791244245889941664376384, 0.46761255787048535049981972889, 2.76373252645181982913087388733, 3.56884299795812426838883866119, 4.96256042832280342802846291119, 6.35021571351565620450459896276, 6.82431008457805971716617235405, 7.52086460923474300785083899871, 8.935029452969523961310575397220, 9.714972325483396845243247558816, 10.80361895638001687435147876425

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