Properties

Label 2-3920-28.27-c1-0-55
Degree $2$
Conductor $3920$
Sign $0.987 + 0.156i$
Analytic cond. $31.3013$
Root an. cond. $5.59476$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.26·3-s + i·5-s + 2.11·9-s − 1.28i·11-s − 2.26i·13-s + 2.26i·15-s + 0.0698i·17-s + 2.44·19-s − 2.21i·23-s − 25-s − 1.99·27-s + 1.98·29-s + 6.31·31-s − 2.89i·33-s + 8.48·37-s + ⋯
L(s)  = 1  + 1.30·3-s + 0.447i·5-s + 0.705·9-s − 0.386i·11-s − 0.627i·13-s + 0.584i·15-s + 0.0169i·17-s + 0.561·19-s − 0.462i·23-s − 0.200·25-s − 0.384·27-s + 0.368·29-s + 1.13·31-s − 0.504i·33-s + 1.39·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 + 0.156i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.987 + 0.156i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3920\)    =    \(2^{4} \cdot 5 \cdot 7^{2}\)
Sign: $0.987 + 0.156i$
Analytic conductor: \(31.3013\)
Root analytic conductor: \(5.59476\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3920} (2351, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3920,\ (\ :1/2),\ 0.987 + 0.156i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.162251365\)
\(L(\frac12)\) \(\approx\) \(3.162251365\)
\(L(\frac{3}{2})\) 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 \( 1 - iT \)
7 \( 1 \)
good3 \( 1 - 2.26T + 3T^{2} \)
11 \( 1 + 1.28iT - 11T^{2} \)
13 \( 1 + 2.26iT - 13T^{2} \)
17 \( 1 - 0.0698iT - 17T^{2} \)
19 \( 1 - 2.44T + 19T^{2} \)
23 \( 1 + 2.21iT - 23T^{2} \)
29 \( 1 - 1.98T + 29T^{2} \)
31 \( 1 - 6.31T + 31T^{2} \)
37 \( 1 - 8.48T + 37T^{2} \)
41 \( 1 + 4.20iT - 41T^{2} \)
43 \( 1 - 5.01iT - 43T^{2} \)
47 \( 1 - 10.6T + 47T^{2} \)
53 \( 1 - 4.01T + 53T^{2} \)
59 \( 1 - 6.89T + 59T^{2} \)
61 \( 1 + 6.73iT - 61T^{2} \)
67 \( 1 + 11.1iT - 67T^{2} \)
71 \( 1 - 14.5iT - 71T^{2} \)
73 \( 1 - 11.8iT - 73T^{2} \)
79 \( 1 + 5.86iT - 79T^{2} \)
83 \( 1 + 5.79T + 83T^{2} \)
89 \( 1 + 17.5iT - 89T^{2} \)
97 \( 1 - 7.06iT - 97T^{2} \)
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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

−8.411914349180189809651498506746, −7.84824519254709671231184333731, −7.21540482485737002436994047050, −6.28877601300773798521735244505, −5.54189842159444337681786019617, −4.44495262606619040548087698954, −3.61866344153615110070556961163, −2.85282419952365344726804947949, −2.34630494111389241214424786964, −0.911623144908698088321576657597, 1.08757916451910323137567681611, 2.19080258429272209636285665774, 2.87609104895970827161920751814, 3.87505823114219802089871588722, 4.47649669151757037278114130655, 5.44323728068235340761396036778, 6.35567716553143790255113430298, 7.31855286189740816959684876107, 7.79545146026235781607292306131, 8.603449148417120921872732243868

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