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Properties

Label 2-552-8.5-c1-0-43
Degree $2$
Conductor $552$
Sign $-0.949 + 0.313i$
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  + (1.14 − 0.832i)2-s − i·3-s + (0.612 − 1.90i)4-s − 1.03i·5-s + (−0.832 − 1.14i)6-s − 3.71·7-s + (−0.885 − 2.68i)8-s − 9-s + (−0.862 − 1.18i)10-s + 1.98i·11-s + (−1.90 − 0.612i)12-s − 4.05i·13-s + (−4.24 + 3.09i)14-s − 1.03·15-s + (−3.24 − 2.33i)16-s + 0.0352·17-s + ⋯
L(s)  = 1  + (0.808 − 0.588i)2-s − 0.577i·3-s + (0.306 − 0.951i)4-s − 0.463i·5-s + (−0.340 − 0.466i)6-s − 1.40·7-s + (−0.313 − 0.949i)8-s − 0.333·9-s + (−0.272 − 0.374i)10-s + 0.597i·11-s + (−0.549 − 0.176i)12-s − 1.12i·13-s + (−1.13 + 0.826i)14-s − 0.267·15-s + (−0.812 − 0.583i)16-s + 0.00855·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.949 + 0.313i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.949 + 0.313i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(552\)    =    \(2^{3} \cdot 3 \cdot 23\)
Sign: $-0.949 + 0.313i$
Analytic conductor: \(4.40774\)
Root analytic conductor: \(2.09946\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{552} (277, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 552,\ (\ :1/2),\ -0.949 + 0.313i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.267657 - 1.66623i\)
\(L(\frac12)\) \(\approx\) \(0.267657 - 1.66623i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.14 + 0.832i)T \)
3 \( 1 + iT \)
23 \( 1 - T \)
good5 \( 1 + 1.03iT - 5T^{2} \)
7 \( 1 + 3.71T + 7T^{2} \)
11 \( 1 - 1.98iT - 11T^{2} \)
13 \( 1 + 4.05iT - 13T^{2} \)
17 \( 1 - 0.0352T + 17T^{2} \)
19 \( 1 - 0.839iT - 19T^{2} \)
29 \( 1 + 8.98iT - 29T^{2} \)
31 \( 1 - 5.65T + 31T^{2} \)
37 \( 1 - 1.63iT - 37T^{2} \)
41 \( 1 + 1.65T + 41T^{2} \)
43 \( 1 - 2.17iT - 43T^{2} \)
47 \( 1 - 9.26T + 47T^{2} \)
53 \( 1 + 9.53iT - 53T^{2} \)
59 \( 1 - 0.521iT - 59T^{2} \)
61 \( 1 - 1.08iT - 61T^{2} \)
67 \( 1 - 3.82iT - 67T^{2} \)
71 \( 1 + 0.0570T + 71T^{2} \)
73 \( 1 - 2.11T + 73T^{2} \)
79 \( 1 + 10.6T + 79T^{2} \)
83 \( 1 + 12.9iT - 83T^{2} \)
89 \( 1 + 8.32T + 89T^{2} \)
97 \( 1 - 17.4T + 97T^{2} \)
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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

−10.27899957116710779940044934660, −9.862390111474163927776772433127, −8.754756677459551814100582685516, −7.48733914607884785205393412049, −6.48379602248750331299020227678, −5.78822684747019396045101453813, −4.65498320773389528787313153788, −3.40126898362482850979704866173, −2.45696218973465971346374066656, −0.73671851182883926614358081645, 2.77936121136013560908421561812, 3.51121997330499279226171360891, 4.55905444467285048497367368163, 5.74563914493580360834131596374, 6.59719803820218203716526685954, 7.17060061414124652651411631057, 8.653240633191354030400753569637, 9.248683724208484336908280280787, 10.41599831917366606228624321868, 11.20074451048449270156174625796

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