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Properties

Label 2-552-23.3-c1-0-9
Degree $2$
Conductor $552$
Sign $0.344 + 0.938i$
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  + (0.415 − 0.909i)3-s + (1.31 − 1.52i)5-s + (4.20 − 2.70i)7-s + (−0.654 − 0.755i)9-s + (−0.430 + 2.99i)11-s + (−3.78 − 2.43i)13-s + (−0.836 − 1.83i)15-s + (7.13 + 2.09i)17-s + (−6.04 + 1.77i)19-s + (−0.711 − 4.95i)21-s + (−4.79 − 0.189i)23-s + (0.135 + 0.939i)25-s + (−0.959 + 0.281i)27-s + (4.74 + 1.39i)29-s + (−2.69 − 5.89i)31-s + ⋯
L(s)  = 1  + (0.239 − 0.525i)3-s + (0.589 − 0.680i)5-s + (1.59 − 1.02i)7-s + (−0.218 − 0.251i)9-s + (−0.129 + 0.902i)11-s + (−1.04 − 0.674i)13-s + (−0.215 − 0.472i)15-s + (1.73 + 0.508i)17-s + (−1.38 + 0.406i)19-s + (−0.155 − 1.08i)21-s + (−0.999 − 0.0394i)23-s + (0.0270 + 0.187i)25-s + (−0.184 + 0.0542i)27-s + (0.881 + 0.258i)29-s + (−0.483 − 1.05i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.344 + 0.938i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.344 + 0.938i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(552\)    =    \(2^{3} \cdot 3 \cdot 23\)
Sign: $0.344 + 0.938i$
Analytic conductor: \(4.40774\)
Root analytic conductor: \(2.09946\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{552} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 552,\ (\ :1/2),\ 0.344 + 0.938i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.56633 - 1.09319i\)
\(L(\frac12)\) \(\approx\) \(1.56633 - 1.09319i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.415 + 0.909i)T \)
23 \( 1 + (4.79 + 0.189i)T \)
good5 \( 1 + (-1.31 + 1.52i)T + (-0.711 - 4.94i)T^{2} \)
7 \( 1 + (-4.20 + 2.70i)T + (2.90 - 6.36i)T^{2} \)
11 \( 1 + (0.430 - 2.99i)T + (-10.5 - 3.09i)T^{2} \)
13 \( 1 + (3.78 + 2.43i)T + (5.40 + 11.8i)T^{2} \)
17 \( 1 + (-7.13 - 2.09i)T + (14.3 + 9.19i)T^{2} \)
19 \( 1 + (6.04 - 1.77i)T + (15.9 - 10.2i)T^{2} \)
29 \( 1 + (-4.74 - 1.39i)T + (24.3 + 15.6i)T^{2} \)
31 \( 1 + (2.69 + 5.89i)T + (-20.3 + 23.4i)T^{2} \)
37 \( 1 + (-1.67 - 1.93i)T + (-5.26 + 36.6i)T^{2} \)
41 \( 1 + (-1.95 + 2.25i)T + (-5.83 - 40.5i)T^{2} \)
43 \( 1 + (1.27 - 2.80i)T + (-28.1 - 32.4i)T^{2} \)
47 \( 1 - 6.39T + 47T^{2} \)
53 \( 1 + (0.0481 - 0.0309i)T + (22.0 - 48.2i)T^{2} \)
59 \( 1 + (5.43 + 3.49i)T + (24.5 + 53.6i)T^{2} \)
61 \( 1 + (-2.41 - 5.28i)T + (-39.9 + 46.1i)T^{2} \)
67 \( 1 + (-0.928 - 6.46i)T + (-64.2 + 18.8i)T^{2} \)
71 \( 1 + (-0.915 - 6.37i)T + (-68.1 + 20.0i)T^{2} \)
73 \( 1 + (10.8 - 3.17i)T + (61.4 - 39.4i)T^{2} \)
79 \( 1 + (12.0 + 7.76i)T + (32.8 + 71.8i)T^{2} \)
83 \( 1 + (-9.00 - 10.3i)T + (-11.8 + 82.1i)T^{2} \)
89 \( 1 + (1.12 - 2.47i)T + (-58.2 - 67.2i)T^{2} \)
97 \( 1 + (-11.8 + 13.6i)T + (-13.8 - 96.0i)T^{2} \)
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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

−10.32278316570595862023932303824, −10.01417308996563920005907783654, −8.616785211306308613081907762531, −7.79566969065521164170500346066, −7.41171864572890294892790317967, −5.88936941335306664397740319837, −4.96894453151625741106929013128, −4.08004441359345364294128007299, −2.19081680124488532927866550047, −1.23204372297305678909935184461, 2.00522415816110040617464152049, 2.88758250046058388106225839529, 4.47647223968058111284796894766, 5.35112588821920111855354921616, 6.18882872108535454581236947163, 7.57422875437135319457901099043, 8.384744529892078246679676324733, 9.149512582583281160604902777535, 10.18651777602269650112947944827, 10.82726164649434178827762442961

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