Properties

Label 2-552-8.5-c1-0-26
Degree $2$
Conductor $552$
Sign $0.546 + 0.837i$
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  + (−0.459 + 1.33i)2-s + i·3-s + (−1.57 − 1.22i)4-s + 2.04i·5-s + (−1.33 − 0.459i)6-s − 4.47·7-s + (2.36 − 1.54i)8-s − 9-s + (−2.74 − 0.941i)10-s − 3.86i·11-s + (1.22 − 1.57i)12-s − 5.65i·13-s + (2.05 − 5.98i)14-s − 2.04·15-s + (0.977 + 3.87i)16-s − 0.763·17-s + ⋯
L(s)  = 1  + (−0.324 + 0.945i)2-s + 0.577i·3-s + (−0.788 − 0.614i)4-s + 0.916i·5-s + (−0.546 − 0.187i)6-s − 1.69·7-s + (0.837 − 0.546i)8-s − 0.333·9-s + (−0.866 − 0.297i)10-s − 1.16i·11-s + (0.354 − 0.455i)12-s − 1.56i·13-s + (0.549 − 1.59i)14-s − 0.529·15-s + (0.244 + 0.969i)16-s − 0.185·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.546 + 0.837i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.546 + 0.837i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(552\)    =    \(2^{3} \cdot 3 \cdot 23\)
Sign: $0.546 + 0.837i$
Analytic conductor: \(4.40774\)
Root analytic conductor: \(2.09946\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{552} (277, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 552,\ (\ :1/2),\ 0.546 + 0.837i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.184554 - 0.0999892i\)
\(L(\frac12)\) \(\approx\) \(0.184554 - 0.0999892i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.459 - 1.33i)T \)
3 \( 1 - iT \)
23 \( 1 - T \)
good5 \( 1 - 2.04iT - 5T^{2} \)
7 \( 1 + 4.47T + 7T^{2} \)
11 \( 1 + 3.86iT - 11T^{2} \)
13 \( 1 + 5.65iT - 13T^{2} \)
17 \( 1 + 0.763T + 17T^{2} \)
19 \( 1 - 5.88iT - 19T^{2} \)
29 \( 1 + 3.85iT - 29T^{2} \)
31 \( 1 + 9.20T + 31T^{2} \)
37 \( 1 + 3.75iT - 37T^{2} \)
41 \( 1 + 3.32T + 41T^{2} \)
43 \( 1 + 1.84iT - 43T^{2} \)
47 \( 1 + 6.55T + 47T^{2} \)
53 \( 1 + 2.26iT - 53T^{2} \)
59 \( 1 + 10.6iT - 59T^{2} \)
61 \( 1 - 9.32iT - 61T^{2} \)
67 \( 1 + 3.64iT - 67T^{2} \)
71 \( 1 + 6.14T + 71T^{2} \)
73 \( 1 + 2.21T + 73T^{2} \)
79 \( 1 + 5.59T + 79T^{2} \)
83 \( 1 + 3.14iT - 83T^{2} \)
89 \( 1 + 0.345T + 89T^{2} \)
97 \( 1 + 19.4T + 97T^{2} \)
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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

−10.33999155512340375394892035599, −9.848835112306715296089139461877, −8.877243410075526637580090215104, −7.950056313102243021041856870579, −6.92964434631457904005950359495, −6.01466962079735728236414729324, −5.55526859945179717254344886041, −3.71379502037302906938324642286, −3.12684761322979933516167243857, −0.13424595055194063385593298722, 1.57986828347929004910421630560, 2.82521362204917698870964097451, 4.11776436974281842527046039202, 5.07234594952395635662197569397, 6.69393926340965436012732398928, 7.22612649054884590050596859210, 8.759945850289063597569671856439, 9.223366085999939567303195962024, 9.807629035379634497734471916184, 11.01135815770940707446846128058

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