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Properties

Label 2-552-69.68-c1-0-13
Degree $2$
Conductor $552$
Sign $0.869 - 0.493i$
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  + (1.72 + 0.121i)3-s + 2.32·5-s + 3.25i·7-s + (2.97 + 0.419i)9-s − 2.65·11-s + 1.62·13-s + (4.01 + 0.282i)15-s − 2.32·17-s + 2.69i·19-s + (−0.395 + 5.62i)21-s + (−2.65 − 3.99i)23-s + 0.403·25-s + (5.08 + 1.08i)27-s − 4.83i·29-s + 0.544·31-s + ⋯
L(s)  = 1  + (0.997 + 0.0700i)3-s + 1.03·5-s + 1.23i·7-s + (0.990 + 0.139i)9-s − 0.799·11-s + 0.450·13-s + (1.03 + 0.0728i)15-s − 0.563·17-s + 0.617i·19-s + (−0.0862 + 1.22i)21-s + (−0.552 − 0.833i)23-s + 0.0807·25-s + (0.977 + 0.208i)27-s − 0.898i·29-s + 0.0977·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.869 - 0.493i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.869 - 0.493i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(552\)    =    \(2^{3} \cdot 3 \cdot 23\)
Sign: $0.869 - 0.493i$
Analytic conductor: \(4.40774\)
Root analytic conductor: \(2.09946\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{552} (137, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 552,\ (\ :1/2),\ 0.869 - 0.493i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.23455 + 0.589194i\)
\(L(\frac12)\) \(\approx\) \(2.23455 + 0.589194i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.72 - 0.121i)T \)
23 \( 1 + (2.65 + 3.99i)T \)
good5 \( 1 - 2.32T + 5T^{2} \)
7 \( 1 - 3.25iT - 7T^{2} \)
11 \( 1 + 2.65T + 11T^{2} \)
13 \( 1 - 1.62T + 13T^{2} \)
17 \( 1 + 2.32T + 17T^{2} \)
19 \( 1 - 2.69iT - 19T^{2} \)
29 \( 1 + 4.83iT - 29T^{2} \)
31 \( 1 - 0.544T + 31T^{2} \)
37 \( 1 + 1.29iT - 37T^{2} \)
41 \( 1 + 5.32iT - 41T^{2} \)
43 \( 1 + 12.6iT - 43T^{2} \)
47 \( 1 - 3.32iT - 47T^{2} \)
53 \( 1 - 12.2T + 53T^{2} \)
59 \( 1 - 13.3iT - 59T^{2} \)
61 \( 1 + 8.82iT - 61T^{2} \)
67 \( 1 - 8.63iT - 67T^{2} \)
71 \( 1 - 1.43iT - 71T^{2} \)
73 \( 1 + 8.85T + 73T^{2} \)
79 \( 1 + 9.85iT - 79T^{2} \)
83 \( 1 + 15.6T + 83T^{2} \)
89 \( 1 - 9.14T + 89T^{2} \)
97 \( 1 - 10.1iT - 97T^{2} \)
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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

−10.46902680883369874151379610213, −9.971661528138905202975946774532, −8.888965499730075946165242321951, −8.556465151475152923175666533972, −7.40230508899826080217284305790, −6.13009526695362346180859792619, −5.43753684742462985102811642952, −4.07267458050412239930120102501, −2.60100469486453286595793139435, −2.03309941461137870412469221080, 1.44444289581607143014595289540, 2.69785554029410178483168830898, 3.87255200450826177318989960059, 4.97821507361863426646966623383, 6.30436915195520730848673803921, 7.22076065363820305224743207898, 8.029145821137161994327910519991, 9.021471610532489228028363768882, 9.881662163896638408000630103316, 10.41027671724170875160746944817

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