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Properties

Label 2-552-24.11-c1-0-20
Degree $2$
Conductor $552$
Sign $0.185 - 0.982i$
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  + (−0.610 − 1.27i)2-s + (0.0769 + 1.73i)3-s + (−1.25 + 1.55i)4-s + 2.14·5-s + (2.16 − 1.15i)6-s + 2.49i·7-s + (2.75 + 0.648i)8-s + (−2.98 + 0.266i)9-s + (−1.30 − 2.73i)10-s + 5.52i·11-s + (−2.79 − 2.05i)12-s − 4.17i·13-s + (3.17 − 1.52i)14-s + (0.164 + 3.70i)15-s + (−0.853 − 3.90i)16-s + 1.36i·17-s + ⋯
L(s)  = 1  + (−0.431 − 0.901i)2-s + (0.0444 + 0.999i)3-s + (−0.627 + 0.778i)4-s + 0.957·5-s + (0.881 − 0.471i)6-s + 0.941i·7-s + (0.973 + 0.229i)8-s + (−0.996 + 0.0887i)9-s + (−0.413 − 0.863i)10-s + 1.66i·11-s + (−0.805 − 0.591i)12-s − 1.15i·13-s + (0.848 − 0.406i)14-s + (0.0425 + 0.956i)15-s + (−0.213 − 0.976i)16-s + 0.331i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.185 - 0.982i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.185 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(552\)    =    \(2^{3} \cdot 3 \cdot 23\)
Sign: $0.185 - 0.982i$
Analytic conductor: \(4.40774\)
Root analytic conductor: \(2.09946\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{552} (323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 552,\ (\ :1/2),\ 0.185 - 0.982i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.834008 + 0.690970i\)
\(L(\frac12)\) \(\approx\) \(0.834008 + 0.690970i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.610 + 1.27i)T \)
3 \( 1 + (-0.0769 - 1.73i)T \)
23 \( 1 + T \)
good5 \( 1 - 2.14T + 5T^{2} \)
7 \( 1 - 2.49iT - 7T^{2} \)
11 \( 1 - 5.52iT - 11T^{2} \)
13 \( 1 + 4.17iT - 13T^{2} \)
17 \( 1 - 1.36iT - 17T^{2} \)
19 \( 1 + 5.17T + 19T^{2} \)
29 \( 1 - 3.61T + 29T^{2} \)
31 \( 1 - 2.18iT - 31T^{2} \)
37 \( 1 - 2.97iT - 37T^{2} \)
41 \( 1 - 8.18iT - 41T^{2} \)
43 \( 1 - 0.482T + 43T^{2} \)
47 \( 1 - 12.8T + 47T^{2} \)
53 \( 1 + 7.36T + 53T^{2} \)
59 \( 1 - 2.97iT - 59T^{2} \)
61 \( 1 - 6.75iT - 61T^{2} \)
67 \( 1 + 5.95T + 67T^{2} \)
71 \( 1 - 5.75T + 71T^{2} \)
73 \( 1 - 12.9T + 73T^{2} \)
79 \( 1 + 7.69iT - 79T^{2} \)
83 \( 1 + 9.34iT - 83T^{2} \)
89 \( 1 + 16.2iT - 89T^{2} \)
97 \( 1 - 6.05T + 97T^{2} \)
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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

−10.48137835511941547824523631090, −10.25580931656283522197892011330, −9.416835245100967942697832575790, −8.755092171273561950609150326450, −7.80721357903607205618670232162, −6.17947536747540617479379559038, −5.14257250031656161942553930875, −4.29478337927381196328761194782, −2.85399845397760251570182450722, −2.02447157599173911238972196773, 0.71344814249312721627378354173, 2.10213988879900463531165538119, 3.98826553556715236732342192004, 5.48679740574217755910494033278, 6.27984251805007166419167449914, 6.82807920285763222885031926750, 7.86267000155058195589655130956, 8.706558362927012732851748730234, 9.391725234198706092171994611459, 10.59226439282357242924156646120

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