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Properties

Label 2-552-184.91-c1-0-1
Degree $2$
Conductor $552$
Sign $-0.671 + 0.741i$
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 − 0.969·5-s + (−1.07 − 0.914i)6-s − 4.55·7-s + (−1.45 + 2.42i)8-s + 9-s + (−1.04 − 0.886i)10-s + 0.915i·11-s + (−0.327 − 1.97i)12-s − 4.49i·13-s + (−4.91 − 4.16i)14-s + 0.969·15-s + (−3.78 + 1.29i)16-s − 5.37i·17-s + ⋯
L(s)  = 1  + (0.762 + 0.646i)2-s − 0.577·3-s + (0.163 + 0.986i)4-s − 0.433·5-s + (−0.440 − 0.373i)6-s − 1.72·7-s + (−0.513 + 0.858i)8-s + 0.333·9-s + (−0.330 − 0.280i)10-s + 0.276i·11-s + (−0.0944 − 0.569i)12-s − 1.24i·13-s + (−1.31 − 1.11i)14-s + 0.250·15-s + (−0.946 + 0.322i)16-s − 1.30i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.671 + 0.741i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.671 + 0.741i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(552\)    =    \(2^{3} \cdot 3 \cdot 23\)
Sign: $-0.671 + 0.741i$
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: \((2,\ 552,\ (\ :1/2),\ -0.671 + 0.741i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0621792 - 0.140153i\)
\(L(\frac12)\) \(\approx\) \(0.0621792 - 0.140153i\)
\(L(\frac{3}{2})\) 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 \( 1 + (4.70 + 0.937i)T \)
good5 \( 1 + 0.969T + 5T^{2} \)
7 \( 1 + 4.55T + 7T^{2} \)
11 \( 1 - 0.915iT - 11T^{2} \)
13 \( 1 + 4.49iT - 13T^{2} \)
17 \( 1 + 5.37iT - 17T^{2} \)
19 \( 1 - 0.688iT - 19T^{2} \)
29 \( 1 - 6.35iT - 29T^{2} \)
31 \( 1 - 5.64iT - 31T^{2} \)
37 \( 1 + 3.25T + 37T^{2} \)
41 \( 1 + 8.87T + 41T^{2} \)
43 \( 1 - 9.54iT - 43T^{2} \)
47 \( 1 - 0.156iT - 47T^{2} \)
53 \( 1 - 2.42T + 53T^{2} \)
59 \( 1 - 3.81T + 59T^{2} \)
61 \( 1 - 12.2T + 61T^{2} \)
67 \( 1 - 10.6iT - 67T^{2} \)
71 \( 1 + 14.9iT - 71T^{2} \)
73 \( 1 + 8.18T + 73T^{2} \)
79 \( 1 + 12.0T + 79T^{2} \)
83 \( 1 + 11.1iT - 83T^{2} \)
89 \( 1 - 10.9iT - 89T^{2} \)
97 \( 1 - 17.8iT - 97T^{2} \)
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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

−11.70522054646743946832015014801, −10.43762278239501163721275907971, −9.681520543439866217910711471086, −8.500505192891498032079234988904, −7.38727852122236537647447055084, −6.75292239689017545915939838465, −5.85143368833109001720095051148, −5.00110684085351676760051838302, −3.69986882581179484746944978630, −2.92282765691012620284705023854, 0.06848537248699147321091933917, 2.10839425107440686801618350123, 3.69092725431107479778728921852, 4.10112039341612885203346602403, 5.70818899970461555594782237408, 6.29271622399503691699743462945, 7.09430579992294037668686515315, 8.679558815089091574975784012154, 9.875203441286347490667638357938, 10.13024023359520921425916753439

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