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Properties

Label 2-3920-35.18-c0-0-2
Degree $2$
Conductor $3920$
Sign $0.954 - 0.299i$
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  + (−0.965 + 0.258i)3-s + (0.965 + 0.258i)5-s + (−0.5 + 0.866i)11-s + (−0.707 − 0.707i)13-s − 15-s + (−0.258 − 0.965i)17-s + (1.22 − 0.707i)19-s + (0.366 − 1.36i)23-s + (0.866 + 0.499i)25-s + (0.707 − 0.707i)27-s + i·29-s + (0.258 − 0.965i)33-s + (1.36 + 0.366i)37-s + (0.866 + 0.500i)39-s + (1 + i)43-s + ⋯
L(s)  = 1  + (−0.965 + 0.258i)3-s + (0.965 + 0.258i)5-s + (−0.5 + 0.866i)11-s + (−0.707 − 0.707i)13-s − 15-s + (−0.258 − 0.965i)17-s + (1.22 − 0.707i)19-s + (0.366 − 1.36i)23-s + (0.866 + 0.499i)25-s + (0.707 − 0.707i)27-s + i·29-s + (0.258 − 0.965i)33-s + (1.36 + 0.366i)37-s + (0.866 + 0.500i)39-s + (1 + i)43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.954 - 0.299i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.954 - 0.299i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3920\)    =    \(2^{4} \cdot 5 \cdot 7^{2}\)
Sign: $0.954 - 0.299i$
Analytic conductor: \(1.95633\)
Root analytic conductor: \(1.39869\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3920} (753, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3920,\ (\ :0),\ 0.954 - 0.299i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.028405837\)
\(L(\frac12)\) \(\approx\) \(1.028405837\)
\(L(1)\) 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 \( 1 + (-0.965 - 0.258i)T \)
7 \( 1 \)
good3 \( 1 + (0.965 - 0.258i)T + (0.866 - 0.5i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + (0.707 + 0.707i)T + iT^{2} \)
17 \( 1 + (0.258 + 0.965i)T + (-0.866 + 0.5i)T^{2} \)
19 \( 1 + (-1.22 + 0.707i)T + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.366 + 1.36i)T + (-0.866 - 0.5i)T^{2} \)
29 \( 1 - iT - T^{2} \)
31 \( 1 + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + (-1.36 - 0.366i)T + (0.866 + 0.5i)T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + (-1 - i)T + iT^{2} \)
47 \( 1 + (-0.965 - 0.258i)T + (0.866 + 0.5i)T^{2} \)
53 \( 1 + (0.866 - 0.5i)T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.866 + 0.5i)T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (0.866 - 0.5i)T^{2} \)
79 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 + (0.5 - 0.866i)T^{2} \)
97 \( 1 + (-0.707 + 0.707i)T - iT^{2} \)
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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

−8.892089327064965614090517918881, −7.73124666627301643773074775431, −7.12631832159370147500245715190, −6.39668477623071163539803935188, −5.57876888759334933553785553340, −4.99653308733591110827748650473, −4.58444505839481332153188096361, −2.80564254438254997010884458091, −2.56001020492088776395240421557, −0.904380664788448816773588719542, 0.927512007171703034816921796165, 2.00114286070444503576532494219, 3.04424797959268466719976104693, 4.15405932296183539434709786894, 5.19406947932008779192279255465, 5.80210022417217658902602612108, 6.03204011669841017562207260081, 7.06596419294540483543503693972, 7.75810907300176144115400586502, 8.720279715380197160830631458135

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