Properties

Label 2-3920-980.939-c0-0-1
Degree $2$
Conductor $3920$
Sign $0.949 + 0.315i$
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  + (0.781 − 0.376i)3-s + (0.900 − 0.433i)5-s + (0.974 + 0.222i)7-s + (−0.153 + 0.193i)9-s + (0.541 − 0.678i)15-s + (0.846 − 0.193i)21-s + (−0.193 + 0.846i)23-s + (0.623 − 0.781i)25-s + (−0.240 + 1.05i)27-s + (0.0990 + 0.433i)29-s + (0.974 − 0.222i)35-s + (1.12 − 0.541i)41-s + (−1.40 − 0.678i)43-s + (−0.0549 + 0.240i)45-s + (−0.541 − 0.678i)47-s + ⋯
L(s)  = 1  + (0.781 − 0.376i)3-s + (0.900 − 0.433i)5-s + (0.974 + 0.222i)7-s + (−0.153 + 0.193i)9-s + (0.541 − 0.678i)15-s + (0.846 − 0.193i)21-s + (−0.193 + 0.846i)23-s + (0.623 − 0.781i)25-s + (−0.240 + 1.05i)27-s + (0.0990 + 0.433i)29-s + (0.974 − 0.222i)35-s + (1.12 − 0.541i)41-s + (−1.40 − 0.678i)43-s + (−0.0549 + 0.240i)45-s + (−0.541 − 0.678i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.949 + 0.315i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.949 + 0.315i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3920\)    =    \(2^{4} \cdot 5 \cdot 7^{2}\)
Sign: $0.949 + 0.315i$
Analytic conductor: \(1.95633\)
Root analytic conductor: \(1.39869\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3920} (1919, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3920,\ (\ :0),\ 0.949 + 0.315i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.172619980\)
\(L(\frac12)\) \(\approx\) \(2.172619980\)
\(L(1)\) 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 \( 1 + (-0.900 + 0.433i)T \)
7 \( 1 + (-0.974 - 0.222i)T \)
good3 \( 1 + (-0.781 + 0.376i)T + (0.623 - 0.781i)T^{2} \)
11 \( 1 + (0.222 - 0.974i)T^{2} \)
13 \( 1 + (0.222 - 0.974i)T^{2} \)
17 \( 1 + (0.900 - 0.433i)T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + (0.193 - 0.846i)T + (-0.900 - 0.433i)T^{2} \)
29 \( 1 + (-0.0990 - 0.433i)T + (-0.900 + 0.433i)T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + (0.900 - 0.433i)T^{2} \)
41 \( 1 + (-1.12 + 0.541i)T + (0.623 - 0.781i)T^{2} \)
43 \( 1 + (1.40 + 0.678i)T + (0.623 + 0.781i)T^{2} \)
47 \( 1 + (0.541 + 0.678i)T + (-0.222 + 0.974i)T^{2} \)
53 \( 1 + (0.900 + 0.433i)T^{2} \)
59 \( 1 + (-0.623 - 0.781i)T^{2} \)
61 \( 1 + (0.400 + 1.75i)T + (-0.900 + 0.433i)T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 + (0.900 + 0.433i)T^{2} \)
73 \( 1 + (0.222 + 0.974i)T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + (1.21 - 1.52i)T + (-0.222 - 0.974i)T^{2} \)
89 \( 1 + (0.277 - 0.347i)T + (-0.222 - 0.974i)T^{2} \)
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

−8.524476360589082793347436511566, −8.061892841089516468779160540236, −7.30644022396604783747140871052, −6.43135026214968401913863390169, −5.37030717292636650115333501018, −5.14871800271878050645648952246, −3.98353365238799565351441544197, −2.89151744844605413215845041845, −2.04208652143771402445343549783, −1.44620562140161034808365075954, 1.38981501605825960274732569610, 2.41974808595187927805432452541, 3.05268449150527049373556350159, 4.12485126140761149069327190549, 4.79867384008359893546821978306, 5.79183853560236324816882207307, 6.40185346222236560801940665356, 7.32118995906159063115277458667, 8.113134248054802747109214884831, 8.672446141628505026316622252460

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