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Properties

Label 2-3920-140.87-c0-0-3
Degree $2$
Conductor $3920$
Sign $0.284 + 0.958i$
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  + (−0.608 + 0.793i)5-s + (0.866 − 0.5i)9-s + (−1.30 − 1.30i)13-s + (−0.198 − 0.739i)17-s + (−0.258 − 0.965i)25-s − 1.41i·29-s + (−0.517 + 1.93i)37-s − 1.84i·41-s + (−0.130 + 0.991i)45-s + (−0.366 − 1.36i)53-s + (0.662 − 0.382i)61-s + (1.83 − 0.241i)65-s + (0.739 − 0.198i)73-s + (0.499 − 0.866i)81-s + (0.707 + 0.292i)85-s + ⋯
L(s)  = 1  + (−0.608 + 0.793i)5-s + (0.866 − 0.5i)9-s + (−1.30 − 1.30i)13-s + (−0.198 − 0.739i)17-s + (−0.258 − 0.965i)25-s − 1.41i·29-s + (−0.517 + 1.93i)37-s − 1.84i·41-s + (−0.130 + 0.991i)45-s + (−0.366 − 1.36i)53-s + (0.662 − 0.382i)61-s + (1.83 − 0.241i)65-s + (0.739 − 0.198i)73-s + (0.499 − 0.866i)81-s + (0.707 + 0.292i)85-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.284 + 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.284 + 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3920\)    =    \(2^{4} \cdot 5 \cdot 7^{2}\)
Sign: $0.284 + 0.958i$
Analytic conductor: \(1.95633\)
Root analytic conductor: \(1.39869\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3920} (3167, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3920,\ (\ :0),\ 0.284 + 0.958i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8943059135\)
\(L(\frac12)\) \(\approx\) \(0.8943059135\)
\(L(1)\) 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 \( 1 + (0.608 - 0.793i)T \)
7 \( 1 \)
good3 \( 1 + (-0.866 + 0.5i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
13 \( 1 + (1.30 + 1.30i)T + iT^{2} \)
17 \( 1 + (0.198 + 0.739i)T + (-0.866 + 0.5i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.866 - 0.5i)T^{2} \)
29 \( 1 + 1.41iT - T^{2} \)
31 \( 1 + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + (0.517 - 1.93i)T + (-0.866 - 0.5i)T^{2} \)
41 \( 1 + 1.84iT - T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 + (-0.866 - 0.5i)T^{2} \)
53 \( 1 + (0.366 + 1.36i)T + (-0.866 + 0.5i)T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (-0.662 + 0.382i)T + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.866 + 0.5i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (-0.739 + 0.198i)T + (0.866 - 0.5i)T^{2} \)
79 \( 1 + (-0.5 + 0.866i)T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 + (-0.382 - 0.662i)T + (-0.5 + 0.866i)T^{2} \)
97 \( 1 + (1.30 - 1.30i)T - iT^{2} \)
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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

−8.233967775076246375225354281640, −7.74693397520141636689738655914, −7.00894108174566357479730724909, −6.56416632512933884838246848902, −5.43768150073276131337388792568, −4.69689093522186642050992803082, −3.80616023736907772205232464956, −3.02837965193462059062018397428, −2.19674684513873403396136706128, −0.51374411589060138566693037368, 1.39682455406892899880959016098, 2.22473271480746796021135779839, 3.56617801428978443922683506468, 4.46405601707140602367830253376, 4.74938273512365047621257575672, 5.72146129057019962580539456823, 6.85996596192851180200347819017, 7.31292841051026243875434092878, 8.004022700590438463929411999559, 8.868038708709022086321428561936

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