Properties

Label 2-3552-296.221-c1-0-74
Degree $2$
Conductor $3552$
Sign $0.0349 - 0.999i$
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 − 2.44·5-s − 4.61·7-s − 9-s − 5.06i·11-s − 0.712·13-s + 2.44i·15-s − 7.27i·17-s + 0.0331·19-s + 4.61i·21-s − 4.39i·23-s + 0.955·25-s + i·27-s − 0.282·29-s − 7.20i·31-s + ⋯
L(s)  = 1  − 0.577i·3-s − 1.09·5-s − 1.74·7-s − 0.333·9-s − 1.52i·11-s − 0.197·13-s + 0.630i·15-s − 1.76i·17-s + 0.00761·19-s + 1.00i·21-s − 0.916i·23-s + 0.191·25-s + 0.192i·27-s − 0.0524·29-s − 1.29i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0349 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0349 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3552\)    =    \(2^{5} \cdot 3 \cdot 37\)
Sign: $0.0349 - 0.999i$
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: \((2,\ 3552,\ (\ :1/2),\ 0.0349 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2375632913\)
\(L(\frac12)\) \(\approx\) \(0.2375632913\)
\(L(\frac{3}{2})\) 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 \( 1 + (-6.04 + 0.650i)T \)
good5 \( 1 + 2.44T + 5T^{2} \)
7 \( 1 + 4.61T + 7T^{2} \)
11 \( 1 + 5.06iT - 11T^{2} \)
13 \( 1 + 0.712T + 13T^{2} \)
17 \( 1 + 7.27iT - 17T^{2} \)
19 \( 1 - 0.0331T + 19T^{2} \)
23 \( 1 + 4.39iT - 23T^{2} \)
29 \( 1 + 0.282T + 29T^{2} \)
31 \( 1 + 7.20iT - 31T^{2} \)
41 \( 1 + 10.2T + 41T^{2} \)
43 \( 1 + 5.99T + 43T^{2} \)
47 \( 1 + 8.51T + 47T^{2} \)
53 \( 1 + 3.34iT - 53T^{2} \)
59 \( 1 - 2.99T + 59T^{2} \)
61 \( 1 + 10.1T + 61T^{2} \)
67 \( 1 - 12.4iT - 67T^{2} \)
71 \( 1 - 12.9T + 71T^{2} \)
73 \( 1 - 5.32T + 73T^{2} \)
79 \( 1 - 1.15iT - 79T^{2} \)
83 \( 1 - 3.97iT - 83T^{2} \)
89 \( 1 + 4.16iT - 89T^{2} \)
97 \( 1 + 13.9iT - 97T^{2} \)
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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

−8.013626050269135550445802034632, −7.18259503635644006855893864912, −6.59556719805481816897781891417, −5.98398476675701560390597798820, −5.00030060042440987643078408319, −3.84103439512579303320578209148, −3.19851102870289218529435367786, −2.59157205547217293611578413975, −0.62627276244444123102376229930, −0.11958759431776183749293982362, 1.79325809830814222482231439528, 3.18787210299769157655147462929, 3.62806336917899091630861293147, 4.37313266397605830158091463447, 5.23213645167889624643370074669, 6.35475317456224183119713370336, 6.79773464618436641416654587210, 7.68714685607032177393346442720, 8.341534430458132980418019711537, 9.292953550926231432789427732495

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