Properties

Label 2-3920-980.179-c0-0-0
Degree $2$
Conductor $3920$
Sign $0.860 + 0.509i$
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  + (−1.29 − 0.400i)3-s + (0.733 − 0.680i)5-s + (0.149 + 0.988i)7-s + (0.702 + 0.479i)9-s + (−1.22 + 0.590i)15-s + (0.202 − 1.34i)21-s + (0.215 − 0.548i)23-s + (0.0747 − 0.997i)25-s + (0.126 + 0.158i)27-s + (0.455 − 0.571i)29-s + (0.781 + 0.623i)35-s + (0.367 + 1.61i)41-s + (0.250 − 1.09i)43-s + (0.841 − 0.126i)45-s + (0.145 + 1.94i)47-s + ⋯
L(s)  = 1  + (−1.29 − 0.400i)3-s + (0.733 − 0.680i)5-s + (0.149 + 0.988i)7-s + (0.702 + 0.479i)9-s + (−1.22 + 0.590i)15-s + (0.202 − 1.34i)21-s + (0.215 − 0.548i)23-s + (0.0747 − 0.997i)25-s + (0.126 + 0.158i)27-s + (0.455 − 0.571i)29-s + (0.781 + 0.623i)35-s + (0.367 + 1.61i)41-s + (0.250 − 1.09i)43-s + (0.841 − 0.126i)45-s + (0.145 + 1.94i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.860 + 0.509i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.860 + 0.509i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3920\)    =    \(2^{4} \cdot 5 \cdot 7^{2}\)
Sign: $0.860 + 0.509i$
Analytic conductor: \(1.95633\)
Root analytic conductor: \(1.39869\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3920} (3119, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3920,\ (\ :0),\ 0.860 + 0.509i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9441785358\)
\(L(\frac12)\) \(\approx\) \(0.9441785358\)
\(L(1)\) 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 \( 1 + (-0.733 + 0.680i)T \)
7 \( 1 + (-0.149 - 0.988i)T \)
good3 \( 1 + (1.29 + 0.400i)T + (0.826 + 0.563i)T^{2} \)
11 \( 1 + (-0.365 + 0.930i)T^{2} \)
13 \( 1 + (-0.623 - 0.781i)T^{2} \)
17 \( 1 + (0.733 - 0.680i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.215 + 0.548i)T + (-0.733 - 0.680i)T^{2} \)
29 \( 1 + (-0.455 + 0.571i)T + (-0.222 - 0.974i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 + (-0.955 - 0.294i)T^{2} \)
41 \( 1 + (-0.367 - 1.61i)T + (-0.900 + 0.433i)T^{2} \)
43 \( 1 + (-0.250 + 1.09i)T + (-0.900 - 0.433i)T^{2} \)
47 \( 1 + (-0.145 - 1.94i)T + (-0.988 + 0.149i)T^{2} \)
53 \( 1 + (-0.955 + 0.294i)T^{2} \)
59 \( 1 + (-0.0747 - 0.997i)T^{2} \)
61 \( 1 + (-1.88 - 0.284i)T + (0.955 + 0.294i)T^{2} \)
67 \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + (0.222 - 0.974i)T^{2} \)
73 \( 1 + (0.988 + 0.149i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (-0.268 + 0.129i)T + (0.623 - 0.781i)T^{2} \)
89 \( 1 + (-0.603 - 0.411i)T + (0.365 + 0.930i)T^{2} \)
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

−8.585988824533035873654298075532, −7.923350712217258755304148191122, −6.79399481111414878091499546458, −6.19734898346170237171040946442, −5.69490145352546708263952874758, −5.03709871028885658586151261163, −4.40566407700974273459588505800, −2.86698921553906637747377270708, −1.92688922137166168500361429676, −0.866500197665559404185357056248, 0.942371759165081753745261452808, 2.21108037965749714794135500965, 3.44077312603175030599951851343, 4.22201115917508334472930044761, 5.23090384641782329696336863750, 5.56632377137183185460220657353, 6.61828746375355697429063844119, 6.89238405435638392968830045592, 7.78348379994622968882589537949, 8.825202293650485360094904696133

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