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Properties

Label 2-552-184.91-c1-0-27
Degree $2$
Conductor $552$
Sign $0.894 - 0.447i$
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  + (−0.695 + 1.23i)2-s + 3-s + (−1.03 − 1.71i)4-s + 1.12·5-s + (−0.695 + 1.23i)6-s + 2.04·7-s + (2.82 − 0.0816i)8-s + 9-s + (−0.783 + 1.38i)10-s − 3.16i·11-s + (−1.03 − 1.71i)12-s + 0.675i·13-s + (−1.41 + 2.51i)14-s + 1.12·15-s + (−1.86 + 3.53i)16-s − 7.33i·17-s + ⋯
L(s)  = 1  + (−0.491 + 0.870i)2-s + 0.577·3-s + (−0.516 − 0.856i)4-s + 0.503·5-s + (−0.283 + 0.502i)6-s + 0.771·7-s + (0.999 − 0.0288i)8-s + 0.333·9-s + (−0.247 + 0.438i)10-s − 0.955i·11-s + (−0.298 − 0.494i)12-s + 0.187i·13-s + (−0.379 + 0.671i)14-s + 0.290·15-s + (−0.466 + 0.884i)16-s − 1.77i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(552\)    =    \(2^{3} \cdot 3 \cdot 23\)
Sign: $0.894 - 0.447i$
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: \((2,\ 552,\ (\ :1/2),\ 0.894 - 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.51992 + 0.359058i\)
\(L(\frac12)\) \(\approx\) \(1.51992 + 0.359058i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.695 - 1.23i)T \)
3 \( 1 - T \)
23 \( 1 + (-4.34 + 2.02i)T \)
good5 \( 1 - 1.12T + 5T^{2} \)
7 \( 1 - 2.04T + 7T^{2} \)
11 \( 1 + 3.16iT - 11T^{2} \)
13 \( 1 - 0.675iT - 13T^{2} \)
17 \( 1 + 7.33iT - 17T^{2} \)
19 \( 1 - 0.636iT - 19T^{2} \)
29 \( 1 - 3.15iT - 29T^{2} \)
31 \( 1 - 8.08iT - 31T^{2} \)
37 \( 1 - 8.85T + 37T^{2} \)
41 \( 1 - 11.0T + 41T^{2} \)
43 \( 1 + 0.447iT - 43T^{2} \)
47 \( 1 - 4.96iT - 47T^{2} \)
53 \( 1 + 12.4T + 53T^{2} \)
59 \( 1 + 7.81T + 59T^{2} \)
61 \( 1 + 0.726T + 61T^{2} \)
67 \( 1 - 14.2iT - 67T^{2} \)
71 \( 1 + 0.0830iT - 71T^{2} \)
73 \( 1 - 1.54T + 73T^{2} \)
79 \( 1 - 6.84T + 79T^{2} \)
83 \( 1 - 8.66iT - 83T^{2} \)
89 \( 1 + 14.5iT - 89T^{2} \)
97 \( 1 + 0.322iT - 97T^{2} \)
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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

−10.74481213192489592808599163579, −9.535141546009195669514328447833, −9.099936387378998364695730552835, −8.153089198355489942807196259718, −7.42304710113042110069241920416, −6.42334223396680783835295414124, −5.36382117127056498096078852693, −4.51577411246989175180126132294, −2.81758606438000496183685089862, −1.21394270518977890618753442492, 1.56229067991056263831665632727, 2.41330224178515804881084762176, 3.86143002121101819526239574593, 4.72168207210770955168411250860, 6.16579336895044653171564586536, 7.73392677622884346540676343303, 7.983661125467878386975612590896, 9.266492881877284384715710243152, 9.699606267055795858139144002694, 10.70874184407438868285971036000

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