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Properties

Label 2-552-24.11-c1-0-63
Degree $2$
Conductor $552$
Sign $0.415 + 0.909i$
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  + (1.29 − 0.568i)2-s + (−1.25 + 1.19i)3-s + (1.35 − 1.47i)4-s + 0.156·5-s + (−0.952 + 2.25i)6-s − 3.17i·7-s + (0.914 − 2.67i)8-s + (0.166 − 2.99i)9-s + (0.202 − 0.0891i)10-s + 3.38i·11-s + (0.0501 + 3.46i)12-s − 3.13i·13-s + (−1.80 − 4.10i)14-s + (−0.197 + 0.186i)15-s + (−0.337 − 3.98i)16-s − 3.67i·17-s + ⋯
L(s)  = 1  + (0.915 − 0.402i)2-s + (−0.726 + 0.687i)3-s + (0.676 − 0.736i)4-s + 0.0700·5-s + (−0.388 + 0.921i)6-s − 1.19i·7-s + (0.323 − 0.946i)8-s + (0.0556 − 0.998i)9-s + (0.0641 − 0.0281i)10-s + 1.02i·11-s + (0.0144 + 0.999i)12-s − 0.868i·13-s + (−0.481 − 1.09i)14-s + (−0.0509 + 0.0481i)15-s + (−0.0844 − 0.996i)16-s − 0.891i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.415 + 0.909i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.415 + 0.909i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(552\)    =    \(2^{3} \cdot 3 \cdot 23\)
Sign: $0.415 + 0.909i$
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: \((2,\ 552,\ (\ :1/2),\ 0.415 + 0.909i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.66198 - 1.06819i\)
\(L(\frac12)\) \(\approx\) \(1.66198 - 1.06819i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.29 + 0.568i)T \)
3 \( 1 + (1.25 - 1.19i)T \)
23 \( 1 - T \)
good5 \( 1 - 0.156T + 5T^{2} \)
7 \( 1 + 3.17iT - 7T^{2} \)
11 \( 1 - 3.38iT - 11T^{2} \)
13 \( 1 + 3.13iT - 13T^{2} \)
17 \( 1 + 3.67iT - 17T^{2} \)
19 \( 1 - 5.24T + 19T^{2} \)
29 \( 1 - 6.81T + 29T^{2} \)
31 \( 1 - 2.63iT - 31T^{2} \)
37 \( 1 + 3.68iT - 37T^{2} \)
41 \( 1 - 3.21iT - 41T^{2} \)
43 \( 1 + 9.14T + 43T^{2} \)
47 \( 1 - 11.1T + 47T^{2} \)
53 \( 1 - 1.24T + 53T^{2} \)
59 \( 1 - 9.90iT - 59T^{2} \)
61 \( 1 + 0.598iT - 61T^{2} \)
67 \( 1 + 6.22T + 67T^{2} \)
71 \( 1 + 11.9T + 71T^{2} \)
73 \( 1 + 8.73T + 73T^{2} \)
79 \( 1 - 12.5iT - 79T^{2} \)
83 \( 1 - 8.47iT - 83T^{2} \)
89 \( 1 + 4.22iT - 89T^{2} \)
97 \( 1 - 9.28T + 97T^{2} \)
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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

−10.45544569954235489673950356211, −10.24169772159845559096037265696, −9.368942949051876774508583847924, −7.51033659713616323819229559909, −6.88500279610535941597796425402, −5.68230810336763581110174650832, −4.86225003171466972268492073947, −4.08184726421333454866328057488, −2.99916029186066991052661723251, −1.00073730731125661681401407378, 1.86519223216327534150753706137, 3.12468321031398977087095527499, 4.60424793515305991109290382880, 5.72631786044123208578733827713, 6.03709591756883343248286777514, 7.07203386847380976368583164436, 8.108804573096732410325203005237, 8.861020710576611698233962187409, 10.35163763681424891788261530492, 11.55960074717297762384815426227

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