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Properties

Label 2-552-184.91-c1-0-11
Degree $2$
Conductor $552$
Sign $0.675 + 0.737i$
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  + (−1.07 − 0.914i)2-s − 3-s + (0.327 + 1.97i)4-s − 3.32·5-s + (1.07 + 0.914i)6-s − 3.62·7-s + (1.45 − 2.42i)8-s + 9-s + (3.58 + 3.04i)10-s + 1.15i·11-s + (−0.327 − 1.97i)12-s + 4.49i·13-s + (3.90 + 3.31i)14-s + 3.32·15-s + (−3.78 + 1.29i)16-s − 4.26i·17-s + ⋯
L(s)  = 1  + (−0.762 − 0.646i)2-s − 0.577·3-s + (0.163 + 0.986i)4-s − 1.48·5-s + (0.440 + 0.373i)6-s − 1.36·7-s + (0.513 − 0.858i)8-s + 0.333·9-s + (1.13 + 0.961i)10-s + 0.347i·11-s + (−0.0944 − 0.569i)12-s + 1.24i·13-s + (1.04 + 0.884i)14-s + 0.858·15-s + (−0.946 + 0.322i)16-s − 1.03i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.675 + 0.737i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.675 + 0.737i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(552\)    =    \(2^{3} \cdot 3 \cdot 23\)
Sign: $0.675 + 0.737i$
Analytic conductor: \(4.40774\)
Root analytic conductor: \(2.09946\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{552} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 552,\ (\ :1/2),\ 0.675 + 0.737i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.342833 - 0.150798i\)
\(L(\frac12)\) \(\approx\) \(0.342833 - 0.150798i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.07 + 0.914i)T \)
3 \( 1 + T \)
23 \( 1 + (-1.37 + 4.59i)T \)
good5 \( 1 + 3.32T + 5T^{2} \)
7 \( 1 + 3.62T + 7T^{2} \)
11 \( 1 - 1.15iT - 11T^{2} \)
13 \( 1 - 4.49iT - 13T^{2} \)
17 \( 1 + 4.26iT - 17T^{2} \)
19 \( 1 + 1.61iT - 19T^{2} \)
29 \( 1 - 1.98iT - 29T^{2} \)
31 \( 1 - 2.69iT - 31T^{2} \)
37 \( 1 - 10.4T + 37T^{2} \)
41 \( 1 - 8.87T + 41T^{2} \)
43 \( 1 - 2.83iT - 43T^{2} \)
47 \( 1 + 13.2iT - 47T^{2} \)
53 \( 1 + 6.02T + 53T^{2} \)
59 \( 1 + 3.81T + 59T^{2} \)
61 \( 1 - 0.0328T + 61T^{2} \)
67 \( 1 + 9.53iT - 67T^{2} \)
71 \( 1 - 7.45iT - 71T^{2} \)
73 \( 1 - 9.56T + 73T^{2} \)
79 \( 1 - 4.25T + 79T^{2} \)
83 \( 1 - 16.3iT - 83T^{2} \)
89 \( 1 - 11.3iT - 89T^{2} \)
97 \( 1 - 8.75iT - 97T^{2} \)
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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

−10.89194787740829799770889303422, −9.683241903048249541615913005059, −9.188025359521722890099694408173, −8.036131184127201398250489553291, −7.02944059864232882993321479604, −6.62155562319011764804486688796, −4.63925657987182823108556362381, −3.82582636235075644126264908301, −2.69959905044963311243474094626, −0.54144489486493034015325294735, 0.67728303059294636469153278108, 3.16453291561035108472814267197, 4.30521428494840218892801448178, 5.78948472271494116538563738713, 6.31014935970978467818238337867, 7.62575360721976441401935567613, 7.87288635226516232591383979806, 9.126356647285576978275809851289, 9.991896324633320269324590175397, 10.84175112315092017574790870864

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