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Properties

Label 2-552-184.91-c1-0-18
Degree $2$
Conductor $552$
Sign $-0.407 + 0.913i$
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  + (−0.409 − 1.35i)2-s − 3-s + (−1.66 + 1.10i)4-s − 3.34·5-s + (0.409 + 1.35i)6-s + 4.25·7-s + (2.18 + 1.79i)8-s + 9-s + (1.37 + 4.52i)10-s + 1.35i·11-s + (1.66 − 1.10i)12-s − 2.36i·13-s + (−1.74 − 5.75i)14-s + 3.34·15-s + (1.53 − 3.69i)16-s + 1.28i·17-s + ⋯
L(s)  = 1  + (−0.289 − 0.957i)2-s − 0.577·3-s + (−0.832 + 0.554i)4-s − 1.49·5-s + (0.167 + 0.552i)6-s + 1.60·7-s + (0.772 + 0.635i)8-s + 0.333·9-s + (0.433 + 1.43i)10-s + 0.409i·11-s + (0.480 − 0.320i)12-s − 0.656i·13-s + (−0.465 − 1.53i)14-s + 0.863·15-s + (0.384 − 0.923i)16-s + 0.312i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.407 + 0.913i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.407 + 0.913i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(552\)    =    \(2^{3} \cdot 3 \cdot 23\)
Sign: $-0.407 + 0.913i$
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: \((2,\ 552,\ (\ :1/2),\ -0.407 + 0.913i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.400400 - 0.617054i\)
\(L(\frac12)\) \(\approx\) \(0.400400 - 0.617054i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.409 + 1.35i)T \)
3 \( 1 + T \)
23 \( 1 + (1.27 + 4.62i)T \)
good5 \( 1 + 3.34T + 5T^{2} \)
7 \( 1 - 4.25T + 7T^{2} \)
11 \( 1 - 1.35iT - 11T^{2} \)
13 \( 1 + 2.36iT - 13T^{2} \)
17 \( 1 - 1.28iT - 17T^{2} \)
19 \( 1 - 1.70iT - 19T^{2} \)
29 \( 1 + 7.24iT - 29T^{2} \)
31 \( 1 + 9.70iT - 31T^{2} \)
37 \( 1 - 3.06T + 37T^{2} \)
41 \( 1 + 2.62T + 41T^{2} \)
43 \( 1 + 10.3iT - 43T^{2} \)
47 \( 1 - 6.05iT - 47T^{2} \)
53 \( 1 - 11.7T + 53T^{2} \)
59 \( 1 - 7.81T + 59T^{2} \)
61 \( 1 - 6.82T + 61T^{2} \)
67 \( 1 - 7.98iT - 67T^{2} \)
71 \( 1 + 13.4iT - 71T^{2} \)
73 \( 1 - 6.03T + 73T^{2} \)
79 \( 1 + 14.0T + 79T^{2} \)
83 \( 1 - 0.0200iT - 83T^{2} \)
89 \( 1 + 2.91iT - 89T^{2} \)
97 \( 1 - 13.4iT - 97T^{2} \)
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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

−10.76458127428804157301431659168, −9.966693520294487692646423437006, −8.503563379750239277340294844976, −8.018899522425658328719815595070, −7.35705366896405217965373511794, −5.54624733796898942896647605323, −4.41475559799431217199630526416, −3.99718517357624639176352174969, −2.23391664484069547603017110513, −0.62501962619615271566617258735, 1.21451008784784854674854590278, 3.80172307411665962304848890537, 4.74999908915888570011544948690, 5.35500580672450336288112687773, 6.85202409757424253367490418883, 7.43429750657491677240966196567, 8.306463363795926884264018001601, 8.852433812197868022274973017241, 10.26542996199122806580766839487, 11.37134686230506915744663656789

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