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Properties

Label 2-3920-140.47-c0-0-3
Degree $2$
Conductor $3920$
Sign $-0.997 + 0.0674i$
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  + (0.130 − 0.991i)5-s + (−0.866 − 0.5i)9-s + (−0.541 − 0.541i)13-s + (−1.78 − 0.478i)17-s + (−0.965 − 0.258i)25-s + 1.41i·29-s + (−1.93 + 0.517i)37-s + 0.765i·41-s + (−0.608 + 0.793i)45-s + (1.36 + 0.366i)53-s + (−1.60 − 0.923i)61-s + (−0.607 + 0.465i)65-s + (0.478 − 1.78i)73-s + (0.499 + 0.866i)81-s + (−0.707 + 1.70i)85-s + ⋯
L(s)  = 1  + (0.130 − 0.991i)5-s + (−0.866 − 0.5i)9-s + (−0.541 − 0.541i)13-s + (−1.78 − 0.478i)17-s + (−0.965 − 0.258i)25-s + 1.41i·29-s + (−1.93 + 0.517i)37-s + 0.765i·41-s + (−0.608 + 0.793i)45-s + (1.36 + 0.366i)53-s + (−1.60 − 0.923i)61-s + (−0.607 + 0.465i)65-s + (0.478 − 1.78i)73-s + (0.499 + 0.866i)81-s + (−0.707 + 1.70i)85-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 + 0.0674i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 + 0.0674i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3920\)    =    \(2^{4} \cdot 5 \cdot 7^{2}\)
Sign: $-0.997 + 0.0674i$
Analytic conductor: \(1.95633\)
Root analytic conductor: \(1.39869\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3920} (607, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3920,\ (\ :0),\ -0.997 + 0.0674i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4170154786\)
\(L(\frac12)\) \(\approx\) \(0.4170154786\)
\(L(1)\) 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 \( 1 + (-0.130 + 0.991i)T \)
7 \( 1 \)
good3 \( 1 + (0.866 + 0.5i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
13 \( 1 + (0.541 + 0.541i)T + iT^{2} \)
17 \( 1 + (1.78 + 0.478i)T + (0.866 + 0.5i)T^{2} \)
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 \( 1 + (1.93 - 0.517i)T + (0.866 - 0.5i)T^{2} \)
41 \( 1 - 0.765iT - T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 + (0.866 - 0.5i)T^{2} \)
53 \( 1 + (-1.36 - 0.366i)T + (0.866 + 0.5i)T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (1.60 + 0.923i)T + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.866 + 0.5i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (-0.478 + 1.78i)T + (-0.866 - 0.5i)T^{2} \)
79 \( 1 + (-0.5 - 0.866i)T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 + (-0.923 + 1.60i)T + (-0.5 - 0.866i)T^{2} \)
97 \( 1 + (0.541 - 0.541i)T - iT^{2} \)
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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

−8.571487292221963607610546420942, −7.61153878669055602052904754317, −6.76550311167504924857608904930, −6.04901600678810999871483439901, −5.10900887955510954605990467813, −4.75422515140150648110470938914, −3.61860791508256109888457532214, −2.72200932428083174981103315663, −1.65924969479245898213628193011, −0.21094263446537228234223033387, 2.11563345474380374881956938979, 2.44066338251405531653884091878, 3.61524590466107907144262445551, 4.39753602341860972543527169078, 5.38450788781303946361307574258, 6.12351620558467718970969699773, 6.84005416794806478157653269173, 7.39485864727103542173747359851, 8.353694151212366352687423471418, 8.906191500491707193605998719443

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