Properties

Label 2-3552-296.221-c1-0-47
Degree $2$
Conductor $3552$
Sign $0.999 + 0.0135i$
Analytic cond. $28.3628$
Root an. cond. $5.32567$
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  − i·3-s + 3.72·5-s + 1.69·7-s − 9-s + 4.68i·11-s + 2.13·13-s − 3.72i·15-s + 2.33i·17-s − 4.49·19-s − 1.69i·21-s − 4.43i·23-s + 8.87·25-s + i·27-s + 2.87·29-s + 5.38i·31-s + ⋯
L(s)  = 1  − 0.577i·3-s + 1.66·5-s + 0.641·7-s − 0.333·9-s + 1.41i·11-s + 0.591·13-s − 0.961i·15-s + 0.566i·17-s − 1.03·19-s − 0.370i·21-s − 0.924i·23-s + 1.77·25-s + 0.192i·27-s + 0.533·29-s + 0.967i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0135i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0135i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3552\)    =    \(2^{5} \cdot 3 \cdot 37\)
Sign: $0.999 + 0.0135i$
Analytic conductor: \(28.3628\)
Root analytic conductor: \(5.32567\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3552} (2737, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3552,\ (\ :1/2),\ 0.999 + 0.0135i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.928105064\)
\(L(\frac12)\) \(\approx\) \(2.928105064\)
\(L(\frac{3}{2})\) 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 \( 1 + iT \)
37 \( 1 + (-1.40 + 5.91i)T \)
good5 \( 1 - 3.72T + 5T^{2} \)
7 \( 1 - 1.69T + 7T^{2} \)
11 \( 1 - 4.68iT - 11T^{2} \)
13 \( 1 - 2.13T + 13T^{2} \)
17 \( 1 - 2.33iT - 17T^{2} \)
19 \( 1 + 4.49T + 19T^{2} \)
23 \( 1 + 4.43iT - 23T^{2} \)
29 \( 1 - 2.87T + 29T^{2} \)
31 \( 1 - 5.38iT - 31T^{2} \)
41 \( 1 - 5.38T + 41T^{2} \)
43 \( 1 + 4.73T + 43T^{2} \)
47 \( 1 - 4.50T + 47T^{2} \)
53 \( 1 - 3.54iT - 53T^{2} \)
59 \( 1 - 12.2T + 59T^{2} \)
61 \( 1 - 13.5T + 61T^{2} \)
67 \( 1 + 2.62iT - 67T^{2} \)
71 \( 1 + 5.83T + 71T^{2} \)
73 \( 1 - 4.35T + 73T^{2} \)
79 \( 1 - 5.01iT - 79T^{2} \)
83 \( 1 + 14.1iT - 83T^{2} \)
89 \( 1 - 16.4iT - 89T^{2} \)
97 \( 1 + 0.171iT - 97T^{2} \)
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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

−8.635393617962173010573681724982, −7.86162260833888966241052283186, −6.80327718966462763010804171792, −6.47968644451712853725810128199, −5.61405383014957437868376000307, −4.91572744088300325692617002563, −4.04060647642736505924183496322, −2.49209703393728852594615450690, −2.03472405619311160483963134714, −1.21569979116466266777226311846, 0.969979513179536941428159694981, 2.06739790271759576683575849331, 2.92302387867103424845584462497, 3.92466696573126158150490388005, 4.93681489376086890683059329747, 5.63607763619551443936094404146, 6.07899240903427678553018598628, 6.86555090062090501017111322171, 8.178176983937600461542838540290, 8.582950742742881106646467838735

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