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Properties

Label 2-552-24.11-c1-0-23
Degree $2$
Conductor $552$
Sign $0.692 + 0.721i$
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  + (−1.40 − 0.136i)2-s + (−1.50 − 0.855i)3-s + (1.96 + 0.382i)4-s − 2.98·5-s + (2.00 + 1.40i)6-s − 1.36i·7-s + (−2.71 − 0.806i)8-s + (1.53 + 2.57i)9-s + (4.19 + 0.405i)10-s + 5.80i·11-s + (−2.62 − 2.25i)12-s + 0.852i·13-s + (−0.185 + 1.91i)14-s + (4.48 + 2.54i)15-s + (3.70 + 1.50i)16-s − 5.38i·17-s + ⋯
L(s)  = 1  + (−0.995 − 0.0961i)2-s + (−0.869 − 0.493i)3-s + (0.981 + 0.191i)4-s − 1.33·5-s + (0.818 + 0.575i)6-s − 0.514i·7-s + (−0.958 − 0.285i)8-s + (0.512 + 0.858i)9-s + (1.32 + 0.128i)10-s + 1.74i·11-s + (−0.759 − 0.651i)12-s + 0.236i·13-s + (−0.0495 + 0.512i)14-s + (1.15 + 0.657i)15-s + (0.926 + 0.375i)16-s − 1.30i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.692 + 0.721i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.692 + 0.721i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(552\)    =    \(2^{3} \cdot 3 \cdot 23\)
Sign: $0.692 + 0.721i$
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: \((2,\ 552,\ (\ :1/2),\ 0.692 + 0.721i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.408581 - 0.174026i\)
\(L(\frac12)\) \(\approx\) \(0.408581 - 0.174026i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.40 + 0.136i)T \)
3 \( 1 + (1.50 + 0.855i)T \)
23 \( 1 - T \)
good5 \( 1 + 2.98T + 5T^{2} \)
7 \( 1 + 1.36iT - 7T^{2} \)
11 \( 1 - 5.80iT - 11T^{2} \)
13 \( 1 - 0.852iT - 13T^{2} \)
17 \( 1 + 5.38iT - 17T^{2} \)
19 \( 1 + 1.59T + 19T^{2} \)
29 \( 1 - 4.01T + 29T^{2} \)
31 \( 1 - 1.32iT - 31T^{2} \)
37 \( 1 + 0.210iT - 37T^{2} \)
41 \( 1 + 10.6iT - 41T^{2} \)
43 \( 1 - 3.80T + 43T^{2} \)
47 \( 1 - 3.46T + 47T^{2} \)
53 \( 1 - 12.5T + 53T^{2} \)
59 \( 1 + 4.46iT - 59T^{2} \)
61 \( 1 + 14.3iT - 61T^{2} \)
67 \( 1 + 9.77T + 67T^{2} \)
71 \( 1 - 3.27T + 71T^{2} \)
73 \( 1 - 2.83T + 73T^{2} \)
79 \( 1 - 4.99iT - 79T^{2} \)
83 \( 1 + 3.09iT - 83T^{2} \)
89 \( 1 - 13.7iT - 89T^{2} \)
97 \( 1 - 10.4T + 97T^{2} \)
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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

−10.69477350700683008128199557760, −9.979354381097275928193756815872, −8.875563552462191512281669354613, −7.61718824069699617975586434814, −7.32571544887352930968676871353, −6.63972629754740965845020791454, −5.02682223271533139072897186082, −4.00284255014651730743039366947, −2.24186478973680754342588419456, −0.61818670872946137127297849860, 0.804909518418301423507020489972, 3.09115339345234298003178582274, 4.14652287174155414002248814638, 5.69900488704226130574231734097, 6.26429596282245106981757079883, 7.48040743472796219261947999482, 8.454921554028299257931998591864, 8.877768332711918514264322953689, 10.30053052605623888869316747705, 10.80724201032096632196292827637

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