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Properties

Label 2-3920-980.599-c0-0-0
Degree $2$
Conductor $3920$
Sign $0.981 - 0.191i$
Analytic cond. $1.95633$
Root an. cond. $1.39869$
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  + (−1.84 + 0.277i)3-s + (−0.365 + 0.930i)5-s + (−0.997 − 0.0747i)7-s + (2.35 − 0.726i)9-s + (0.414 − 1.81i)15-s + (1.85 − 0.139i)21-s + (0.246 + 0.167i)23-s + (−0.733 − 0.680i)25-s + (−2.45 + 1.18i)27-s + (−1.48 − 0.716i)29-s + (0.433 − 0.900i)35-s + (−1.19 − 1.49i)41-s + (0.367 − 0.460i)43-s + (−0.184 + 2.45i)45-s + (1.14 − 1.06i)47-s + ⋯
L(s)  = 1  + (−1.84 + 0.277i)3-s + (−0.365 + 0.930i)5-s + (−0.997 − 0.0747i)7-s + (2.35 − 0.726i)9-s + (0.414 − 1.81i)15-s + (1.85 − 0.139i)21-s + (0.246 + 0.167i)23-s + (−0.733 − 0.680i)25-s + (−2.45 + 1.18i)27-s + (−1.48 − 0.716i)29-s + (0.433 − 0.900i)35-s + (−1.19 − 1.49i)41-s + (0.367 − 0.460i)43-s + (−0.184 + 2.45i)45-s + (1.14 − 1.06i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 - 0.191i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 - 0.191i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3920\)    =    \(2^{4} \cdot 5 \cdot 7^{2}\)
Sign: $0.981 - 0.191i$
Analytic conductor: \(1.95633\)
Root analytic conductor: \(1.39869\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3920} (2559, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3920,\ (\ :0),\ 0.981 - 0.191i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4163892318\)
\(L(\frac12)\) \(\approx\) \(0.4163892318\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (0.365 - 0.930i)T \)
7 \( 1 + (0.997 + 0.0747i)T \)
good3 \( 1 + (1.84 - 0.277i)T + (0.955 - 0.294i)T^{2} \)
11 \( 1 + (-0.826 - 0.563i)T^{2} \)
13 \( 1 + (0.900 - 0.433i)T^{2} \)
17 \( 1 + (-0.365 + 0.930i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.246 - 0.167i)T + (0.365 + 0.930i)T^{2} \)
29 \( 1 + (1.48 + 0.716i)T + (0.623 + 0.781i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 + (0.988 - 0.149i)T^{2} \)
41 \( 1 + (1.19 + 1.49i)T + (-0.222 + 0.974i)T^{2} \)
43 \( 1 + (-0.367 + 0.460i)T + (-0.222 - 0.974i)T^{2} \)
47 \( 1 + (-1.14 + 1.06i)T + (0.0747 - 0.997i)T^{2} \)
53 \( 1 + (0.988 + 0.149i)T^{2} \)
59 \( 1 + (0.733 - 0.680i)T^{2} \)
61 \( 1 + (-0.147 - 1.97i)T + (-0.988 + 0.149i)T^{2} \)
67 \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + (-0.623 + 0.781i)T^{2} \)
73 \( 1 + (-0.0747 - 0.997i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.443 - 1.94i)T + (-0.900 - 0.433i)T^{2} \)
89 \( 1 + (-1.57 + 0.487i)T + (0.826 - 0.563i)T^{2} \)
97 \( 1 - T^{2} \)
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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

−8.824465154134997857764084468129, −7.38252027212965335367703486185, −7.15329433268058053623032698793, −6.33833367284194179412176150600, −5.82207195376269756353612544801, −5.13180926611693597738633441222, −3.98193128977132501441951397997, −3.59506727183809977794778990193, −2.19145983446494897149947968336, −0.51425805839206165169341639832, 0.68182760924923101526484656971, 1.75085863789873871499104083191, 3.37039512782331414015471118357, 4.33980463236487640452715904694, 4.99473676718593618868296720790, 5.71302720083778479938689211440, 6.24692292872595640942666933132, 7.04632543131682898248982168963, 7.62255828019277694473180504765, 8.662622604795692663338369741800

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