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Properties

Label 2-552-23.12-c1-0-6
Degree $2$
Conductor $552$
Sign $0.910 + 0.413i$
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  + (0.142 − 0.989i)3-s + (3.49 − 1.02i)5-s + (1.49 + 1.72i)7-s + (−0.959 − 0.281i)9-s + (2.79 + 1.79i)11-s + (0.353 − 0.407i)13-s + (−0.519 − 3.60i)15-s + (−0.777 − 1.70i)17-s + (−2.95 + 6.45i)19-s + (1.92 − 1.23i)21-s + (−2.42 + 4.14i)23-s + (6.98 − 4.48i)25-s + (−0.415 + 0.909i)27-s + (−1.60 − 3.51i)29-s + (−1.07 − 7.47i)31-s + ⋯
L(s)  = 1  + (0.0821 − 0.571i)3-s + (1.56 − 0.459i)5-s + (0.565 + 0.652i)7-s + (−0.319 − 0.0939i)9-s + (0.842 + 0.541i)11-s + (0.0980 − 0.113i)13-s + (−0.134 − 0.932i)15-s + (−0.188 − 0.412i)17-s + (−0.676 + 1.48i)19-s + (0.419 − 0.269i)21-s + (−0.504 + 0.863i)23-s + (1.39 − 0.897i)25-s + (−0.0799 + 0.175i)27-s + (−0.298 − 0.653i)29-s + (−0.192 − 1.34i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.910 + 0.413i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.910 + 0.413i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(552\)    =    \(2^{3} \cdot 3 \cdot 23\)
Sign: $0.910 + 0.413i$
Analytic conductor: \(4.40774\)
Root analytic conductor: \(2.09946\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{552} (265, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 552,\ (\ :1/2),\ 0.910 + 0.413i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.98956 - 0.431109i\)
\(L(\frac12)\) \(\approx\) \(1.98956 - 0.431109i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.142 + 0.989i)T \)
23 \( 1 + (2.42 - 4.14i)T \)
good5 \( 1 + (-3.49 + 1.02i)T + (4.20 - 2.70i)T^{2} \)
7 \( 1 + (-1.49 - 1.72i)T + (-0.996 + 6.92i)T^{2} \)
11 \( 1 + (-2.79 - 1.79i)T + (4.56 + 10.0i)T^{2} \)
13 \( 1 + (-0.353 + 0.407i)T + (-1.85 - 12.8i)T^{2} \)
17 \( 1 + (0.777 + 1.70i)T + (-11.1 + 12.8i)T^{2} \)
19 \( 1 + (2.95 - 6.45i)T + (-12.4 - 14.3i)T^{2} \)
29 \( 1 + (1.60 + 3.51i)T + (-18.9 + 21.9i)T^{2} \)
31 \( 1 + (1.07 + 7.47i)T + (-29.7 + 8.73i)T^{2} \)
37 \( 1 + (8.13 + 2.38i)T + (31.1 + 20.0i)T^{2} \)
41 \( 1 + (-1.10 + 0.323i)T + (34.4 - 22.1i)T^{2} \)
43 \( 1 + (-1.30 + 9.10i)T + (-41.2 - 12.1i)T^{2} \)
47 \( 1 - 2.73T + 47T^{2} \)
53 \( 1 + (-5.08 - 5.86i)T + (-7.54 + 52.4i)T^{2} \)
59 \( 1 + (2.77 - 3.20i)T + (-8.39 - 58.3i)T^{2} \)
61 \( 1 + (-0.208 - 1.44i)T + (-58.5 + 17.1i)T^{2} \)
67 \( 1 + (4.35 - 2.79i)T + (27.8 - 60.9i)T^{2} \)
71 \( 1 + (1.92 - 1.23i)T + (29.4 - 64.5i)T^{2} \)
73 \( 1 + (-6.68 + 14.6i)T + (-47.8 - 55.1i)T^{2} \)
79 \( 1 + (4.38 - 5.06i)T + (-11.2 - 78.1i)T^{2} \)
83 \( 1 + (15.2 + 4.46i)T + (69.8 + 44.8i)T^{2} \)
89 \( 1 + (2.52 - 17.5i)T + (-85.3 - 25.0i)T^{2} \)
97 \( 1 + (10.3 - 3.02i)T + (81.6 - 52.4i)T^{2} \)
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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

−10.61060915049791891781152760697, −9.663236289957645822196016147798, −9.059008913578792813681799053364, −8.177745267685374962762532525839, −7.04118464767362091542273036240, −5.87051693143665847985217547846, −5.56492570124673525973876213147, −4.08147023218904026580154419151, −2.21191268005232786024424029488, −1.62858820877639421705269224585, 1.58883547298142780032290599395, 2.90201682114475429871125055511, 4.24528997549373715219126485035, 5.24499068628011941578859045356, 6.33918292193598497375723916609, 6.94642896480147581749870128877, 8.538982865612128486121407867502, 9.095271786460268072103426019711, 10.09547040882235825529968638654, 10.72595123184663976797315822816

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