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Properties

Label 2-552-69.68-c1-0-3
Degree $2$
Conductor $552$
Sign $-0.119 - 0.992i$
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  + (−1.61 − 0.618i)3-s + 0.874·5-s + 1.41i·7-s + (2.23 + 2.00i)9-s − 5.11·11-s − 3.23·13-s + (−1.41 − 0.540i)15-s + 4.91·17-s + 8.27i·19-s + (0.874 − 2.28i)21-s + (4.24 − 2.23i)23-s − 4.23·25-s + (−2.38 − 4.61i)27-s + 5.23i·29-s − 6·31-s + ⋯
L(s)  = 1  + (−0.934 − 0.356i)3-s + 0.390·5-s + 0.534i·7-s + (0.745 + 0.666i)9-s − 1.54·11-s − 0.897·13-s + (−0.365 − 0.139i)15-s + 1.19·17-s + 1.89i·19-s + (0.190 − 0.499i)21-s + (0.884 − 0.466i)23-s − 0.847·25-s + (−0.458 − 0.888i)27-s + 0.972i·29-s − 1.07·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.119 - 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.119 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(552\)    =    \(2^{3} \cdot 3 \cdot 23\)
Sign: $-0.119 - 0.992i$
Analytic conductor: \(4.40774\)
Root analytic conductor: \(2.09946\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{552} (137, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 552,\ (\ :1/2),\ -0.119 - 0.992i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.454397 + 0.512576i\)
\(L(\frac12)\) \(\approx\) \(0.454397 + 0.512576i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (1.61 + 0.618i)T \)
23 \( 1 + (-4.24 + 2.23i)T \)
good5 \( 1 - 0.874T + 5T^{2} \)
7 \( 1 - 1.41iT - 7T^{2} \)
11 \( 1 + 5.11T + 11T^{2} \)
13 \( 1 + 3.23T + 13T^{2} \)
17 \( 1 - 4.91T + 17T^{2} \)
19 \( 1 - 8.27iT - 19T^{2} \)
29 \( 1 - 5.23iT - 29T^{2} \)
31 \( 1 + 6T + 31T^{2} \)
37 \( 1 - 10.7iT - 37T^{2} \)
41 \( 1 + 0.472iT - 41T^{2} \)
43 \( 1 - 10.0iT - 43T^{2} \)
47 \( 1 - 11.7iT - 47T^{2} \)
53 \( 1 + 0.874T + 53T^{2} \)
59 \( 1 + 8iT - 59T^{2} \)
61 \( 1 + 1.20iT - 61T^{2} \)
67 \( 1 + 6.53iT - 67T^{2} \)
71 \( 1 + 12.1iT - 71T^{2} \)
73 \( 1 + 7.23T + 73T^{2} \)
79 \( 1 + 6.65iT - 79T^{2} \)
83 \( 1 - 7.94T + 83T^{2} \)
89 \( 1 + 0.333T + 89T^{2} \)
97 \( 1 + 2.41iT - 97T^{2} \)
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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

−10.92256762557972590683132168324, −10.22851633947034175213596919118, −9.568092889679902847024415154516, −8.009559843325670944722297739356, −7.57828592748907092648627483451, −6.24219190078731998066759089410, −5.49783778410619329251763567140, −4.84832393844643329444035608746, −3.03413336912760846863473329961, −1.66714262347443366177864946724, 0.43322026802330067264163923141, 2.47288230145750510600577473624, 3.96538771151568190099989655306, 5.28032730386854641870035321553, 5.49905719901777442993436901541, 7.10946674041823916608311214475, 7.48227717778776942478551029965, 9.036646765308494199899323183374, 9.935060798973830572294774855762, 10.49465801450100385164659539394

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