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Properties

Label 2-552-8.5-c1-0-38
Degree $2$
Conductor $552$
Sign $-0.966 + 0.255i$
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  + (−1.15 + 0.809i)2-s − i·3-s + (0.688 − 1.87i)4-s − 3.56i·5-s + (0.809 + 1.15i)6-s − 2.96·7-s + (0.723 + 2.73i)8-s − 9-s + (2.88 + 4.13i)10-s − 2.43i·11-s + (−1.87 − 0.688i)12-s − 2.03i·13-s + (3.43 − 2.39i)14-s − 3.56·15-s + (−3.05 − 2.58i)16-s + 2.92·17-s + ⋯
L(s)  = 1  + (−0.819 + 0.572i)2-s − 0.577i·3-s + (0.344 − 0.938i)4-s − 1.59i·5-s + (0.330 + 0.473i)6-s − 1.11·7-s + (0.255 + 0.966i)8-s − 0.333·9-s + (0.912 + 1.30i)10-s − 0.734i·11-s + (−0.542 − 0.198i)12-s − 0.565i·13-s + (0.917 − 0.641i)14-s − 0.920·15-s + (−0.763 − 0.646i)16-s + 0.708·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.966 + 0.255i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.966 + 0.255i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(552\)    =    \(2^{3} \cdot 3 \cdot 23\)
Sign: $-0.966 + 0.255i$
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: \((2,\ 552,\ (\ :1/2),\ -0.966 + 0.255i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0603714 - 0.464454i\)
\(L(\frac12)\) \(\approx\) \(0.0603714 - 0.464454i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.15 - 0.809i)T \)
3 \( 1 + iT \)
23 \( 1 + T \)
good5 \( 1 + 3.56iT - 5T^{2} \)
7 \( 1 + 2.96T + 7T^{2} \)
11 \( 1 + 2.43iT - 11T^{2} \)
13 \( 1 + 2.03iT - 13T^{2} \)
17 \( 1 - 2.92T + 17T^{2} \)
19 \( 1 - 5.23iT - 19T^{2} \)
29 \( 1 + 0.531iT - 29T^{2} \)
31 \( 1 + 10.1T + 31T^{2} \)
37 \( 1 - 9.71iT - 37T^{2} \)
41 \( 1 + 3.68T + 41T^{2} \)
43 \( 1 - 1.24iT - 43T^{2} \)
47 \( 1 - 4.08T + 47T^{2} \)
53 \( 1 + 13.3iT - 53T^{2} \)
59 \( 1 + 9.85iT - 59T^{2} \)
61 \( 1 + 14.6iT - 61T^{2} \)
67 \( 1 - 3.93iT - 67T^{2} \)
71 \( 1 - 5.93T + 71T^{2} \)
73 \( 1 - 3.85T + 73T^{2} \)
79 \( 1 + 7.13T + 79T^{2} \)
83 \( 1 - 13.1iT - 83T^{2} \)
89 \( 1 + 10.0T + 89T^{2} \)
97 \( 1 - 9.43T + 97T^{2} \)
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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

−9.944709215663275906454017000736, −9.459199902352580350904085847793, −8.335390405522785768368658574661, −8.087746447945451560464831630053, −6.78709298369836260284307688412, −5.82149040710631009124819728754, −5.23282242336576598901795169113, −3.47016929802716030330375857000, −1.58716066447459283511759234603, −0.34656463063553935943942988897, 2.34479173934434026527759648796, 3.21264505899805247446163058143, 4.07933270778670240248284961612, 5.93870190411310104464517937663, 7.08727477175963725505401915409, 7.33033167856194990370445673289, 9.016071559884061424983776724707, 9.526304609370978868626211118495, 10.39804117922098707163770781777, 10.81990300674270682935400037064

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