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Properties

Label 2-552-24.11-c1-0-59
Degree $2$
Conductor $552$
Sign $0.974 - 0.225i$
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 + (−2.23 + 0.998i)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 + (−3.28 + 1.10i)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 + (−0.913 + 0.407i)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 + (−0.948 + 0.317i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.974 - 0.225i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.974 - 0.225i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(552\)    =    \(2^{3} \cdot 3 \cdot 23\)
Sign: $0.974 - 0.225i$
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: \((2,\ 552,\ (\ :1/2),\ 0.974 - 0.225i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.55786 + 0.291927i\)
\(L(\frac12)\) \(\approx\) \(2.55786 + 0.291927i\)
\(L(\frac{3}{2})\) 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

−10.85257827849628973782041619558, −10.34000938371422402858043123211, −9.278576078145035321079132207328, −8.029887289655981319558313237273, −6.51645644539436947245489592998, −6.10241367636708038609666956410, −5.42445266016549534336853005253, −4.26791921279278199945336763543, −3.24232669987694057775930252550, −1.53576241263429411502496925809, 1.76616295738714913517464708486, 2.47918842060382858962601907998, 4.47187661901743416934261938421, 5.29962243060748896617550983071, 5.92882613993413394925286970320, 6.89262983512111896521320603984, 7.53745110150066735458292538218, 9.354948375011133055148047466819, 10.07160597667179348300303537128, 10.86924134018738681029122558951

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