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Properties

Label 2-2400-600.221-c0-0-0
Degree $2$
Conductor $2400$
Sign $0.968 - 0.248i$
Analytic cond. $1.19775$
Root an. cond. $1.09442$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.809 − 0.587i)3-s + (−0.309 + 0.951i)5-s + 1.61·7-s + (0.309 + 0.951i)9-s + (0.190 − 0.587i)11-s + (0.809 − 0.587i)15-s + (−1.30 − 0.951i)21-s + (−0.809 − 0.587i)25-s + (0.309 − 0.951i)27-s + (1.61 + 1.17i)29-s + (−1.30 + 0.951i)31-s + (−0.5 + 0.363i)33-s + (−0.500 + 1.53i)35-s − 45-s + 1.61·49-s + ⋯
L(s)  = 1  + (−0.809 − 0.587i)3-s + (−0.309 + 0.951i)5-s + 1.61·7-s + (0.309 + 0.951i)9-s + (0.190 − 0.587i)11-s + (0.809 − 0.587i)15-s + (−1.30 − 0.951i)21-s + (−0.809 − 0.587i)25-s + (0.309 − 0.951i)27-s + (1.61 + 1.17i)29-s + (−1.30 + 0.951i)31-s + (−0.5 + 0.363i)33-s + (−0.500 + 1.53i)35-s − 45-s + 1.61·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 - 0.248i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 - 0.248i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2400\)    =    \(2^{5} \cdot 3 \cdot 5^{2}\)
Sign: $0.968 - 0.248i$
Analytic conductor: \(1.19775\)
Root analytic conductor: \(1.09442\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2400} (2321, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2400,\ (\ :0),\ 0.968 - 0.248i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.053284487\)
\(L(\frac12)\) \(\approx\) \(1.053284487\)
\(L(1)\) 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 \( 1 + (0.809 + 0.587i)T \)
5 \( 1 + (0.309 - 0.951i)T \)
good7 \( 1 - 1.61T + T^{2} \)
11 \( 1 + (-0.190 + 0.587i)T + (-0.809 - 0.587i)T^{2} \)
13 \( 1 + (0.809 - 0.587i)T^{2} \)
17 \( 1 + (-0.309 + 0.951i)T^{2} \)
19 \( 1 + (-0.309 + 0.951i)T^{2} \)
23 \( 1 + (0.809 + 0.587i)T^{2} \)
29 \( 1 + (-1.61 - 1.17i)T + (0.309 + 0.951i)T^{2} \)
31 \( 1 + (1.30 - 0.951i)T + (0.309 - 0.951i)T^{2} \)
37 \( 1 + (0.809 - 0.587i)T^{2} \)
41 \( 1 + (0.809 - 0.587i)T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (-0.309 - 0.951i)T^{2} \)
53 \( 1 + (-0.5 - 0.363i)T + (0.309 + 0.951i)T^{2} \)
59 \( 1 + (-0.190 - 0.587i)T + (-0.809 + 0.587i)T^{2} \)
61 \( 1 + (0.809 + 0.587i)T^{2} \)
67 \( 1 + (-0.309 + 0.951i)T^{2} \)
71 \( 1 + (-0.309 - 0.951i)T^{2} \)
73 \( 1 + (-0.618 + 1.90i)T + (-0.809 - 0.587i)T^{2} \)
79 \( 1 + (-0.5 - 0.363i)T + (0.309 + 0.951i)T^{2} \)
83 \( 1 + (-1.30 + 0.951i)T + (0.309 - 0.951i)T^{2} \)
89 \( 1 + (0.809 + 0.587i)T^{2} \)
97 \( 1 + (-1.30 - 0.951i)T + (0.309 + 0.951i)T^{2} \)
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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

−8.954733257130962132168270505949, −8.214890825042618871100685737178, −7.56195530570196327839195622082, −6.92107308205558999857001883016, −6.15112086203231042651184778303, −5.24677071961994349401372056339, −4.61495433540472036230850692659, −3.44563282031739004739689433185, −2.24706205346856721403883163236, −1.23607732875796271055703332607, 0.981636270830028154117886976444, 2.09489177343185818979771367631, 3.88265088925148153961162139280, 4.45531779512726585346038804338, 5.07121611629041715810217304114, 5.69284485194995060430470376075, 6.77709682716894868070903531113, 7.76249488009496369146046727752, 8.313068427677982913253053276519, 9.161606368855846043832927482732

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