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Properties

Label 2-2352-4.3-c2-0-56
Degree $2$
Conductor $2352$
Sign $0.866 - 0.5i$
Analytic cond. $64.0873$
Root an. cond. $8.00545$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.73i·3-s + 9.08·5-s − 2.99·9-s + 8.80i·11-s + 15.7i·15-s − 2.91·17-s − 17.6i·19-s − 29.5i·23-s + 57.4·25-s − 5.19i·27-s + 48.4·29-s − 38.3i·31-s − 15.2·33-s − 20.4·37-s + 26.9·41-s + ⋯
L(s)  = 1  + 0.577i·3-s + 1.81·5-s − 0.333·9-s + 0.800i·11-s + 1.04i·15-s − 0.171·17-s − 0.926i·19-s − 1.28i·23-s + 2.29·25-s − 0.192i·27-s + 1.67·29-s − 1.23i·31-s − 0.462·33-s − 0.553·37-s + 0.656·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.866 - 0.5i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.866 - 0.5i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2352\)    =    \(2^{4} \cdot 3 \cdot 7^{2}\)
Sign: $0.866 - 0.5i$
Analytic conductor: \(64.0873\)
Root analytic conductor: \(8.00545\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{2352} (1471, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2352,\ (\ :1),\ 0.866 - 0.5i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(3.194881469\)
\(L(\frac12)\) \(\approx\) \(3.194881469\)
\(L(2)\) 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 \( 1 - 1.73iT \)
7 \( 1 \)
good5 \( 1 - 9.08T + 25T^{2} \)
11 \( 1 - 8.80iT - 121T^{2} \)
13 \( 1 + 169T^{2} \)
17 \( 1 + 2.91T + 289T^{2} \)
19 \( 1 + 17.6iT - 361T^{2} \)
23 \( 1 + 29.5iT - 529T^{2} \)
29 \( 1 - 48.4T + 841T^{2} \)
31 \( 1 + 38.3iT - 961T^{2} \)
37 \( 1 + 20.4T + 1.36e3T^{2} \)
41 \( 1 - 26.9T + 1.68e3T^{2} \)
43 \( 1 - 27.7iT - 1.84e3T^{2} \)
47 \( 1 - 62.9iT - 2.20e3T^{2} \)
53 \( 1 - 84.4T + 2.80e3T^{2} \)
59 \( 1 - 42.1iT - 3.48e3T^{2} \)
61 \( 1 - 102.T + 3.72e3T^{2} \)
67 \( 1 - 52.8iT - 4.48e3T^{2} \)
71 \( 1 - 64.8iT - 5.04e3T^{2} \)
73 \( 1 - 30.4T + 5.32e3T^{2} \)
79 \( 1 + 149. iT - 6.24e3T^{2} \)
83 \( 1 + 0.573iT - 6.88e3T^{2} \)
89 \( 1 + 10.0T + 7.92e3T^{2} \)
97 \( 1 - 113.T + 9.40e3T^{2} \)
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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

−9.049284584478879108526493020338, −8.382905777746486310028582936015, −7.10438232413911326694825514617, −6.43049615526932056643696853623, −5.74376733350372437952058023843, −4.87603016350702272286427281518, −4.29782674744180591255982807253, −2.70222769996513676642567885309, −2.31215675535558718170582129274, −0.959590973427468262272272259386, 0.969271288031785980815988586064, 1.81098434828358126156703514361, 2.67876800519266624528750729045, 3.67754658176515912515547159333, 5.27215587215522862712307775107, 5.51543927383642575911811987539, 6.47577482355523032252433180077, 6.89836176919352735865100332386, 8.126117658197395932296657226387, 8.738814353789604425621662273844

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