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Properties

Label 1-3920-3920.83-r1-0-0
Degree $1$
Conductor $3920$
Sign $-0.517 + 0.855i$
Analytic cond. $421.262$
Root an. cond. $421.262$
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  + (0.222 + 0.974i)3-s + (−0.900 + 0.433i)9-s + (0.433 − 0.900i)11-s + (−0.900 − 0.433i)13-s + (0.781 + 0.623i)17-s − i·19-s + (−0.781 + 0.623i)23-s + (−0.623 − 0.781i)27-s + (−0.781 − 0.623i)29-s + 31-s + (0.974 + 0.222i)33-s + (−0.623 + 0.781i)37-s + (0.222 − 0.974i)39-s + (−0.222 − 0.974i)41-s + (0.222 − 0.974i)43-s + ⋯
L(s)  = 1  + (0.222 + 0.974i)3-s + (−0.900 + 0.433i)9-s + (0.433 − 0.900i)11-s + (−0.900 − 0.433i)13-s + (0.781 + 0.623i)17-s − i·19-s + (−0.781 + 0.623i)23-s + (−0.623 − 0.781i)27-s + (−0.781 − 0.623i)29-s + 31-s + (0.974 + 0.222i)33-s + (−0.623 + 0.781i)37-s + (0.222 − 0.974i)39-s + (−0.222 − 0.974i)41-s + (0.222 − 0.974i)43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.517 + 0.855i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.517 + 0.855i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3920\)    =    \(2^{4} \cdot 5 \cdot 7^{2}\)
Sign: $-0.517 + 0.855i$
Analytic conductor: \(421.262\)
Root analytic conductor: \(421.262\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3920} (83, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3920,\ (1:\ ),\ -0.517 + 0.855i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6080897783 + 1.078980496i\)
\(L(\frac12)\) \(\approx\) \(0.6080897783 + 1.078980496i\)
\(L(1)\) \(\approx\) \(0.9549032164 + 0.2545091124i\)
\(L(1)\) \(\approx\) \(0.9549032164 + 0.2545091124i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good3 \( 1 + (0.222 + 0.974i)T \)
11 \( 1 + (0.433 - 0.900i)T \)
13 \( 1 + (-0.900 - 0.433i)T \)
17 \( 1 + (0.781 + 0.623i)T \)
19 \( 1 - iT \)
23 \( 1 + (-0.781 + 0.623i)T \)
29 \( 1 + (-0.781 - 0.623i)T \)
31 \( 1 + T \)
37 \( 1 + (-0.623 + 0.781i)T \)
41 \( 1 + (-0.222 - 0.974i)T \)
43 \( 1 + (0.222 - 0.974i)T \)
47 \( 1 + (-0.433 + 0.900i)T \)
53 \( 1 + (0.623 + 0.781i)T \)
59 \( 1 + (-0.974 - 0.222i)T \)
61 \( 1 + (-0.781 - 0.623i)T \)
67 \( 1 - T \)
71 \( 1 + (0.623 + 0.781i)T \)
73 \( 1 + (-0.433 - 0.900i)T \)
79 \( 1 + T \)
83 \( 1 + (0.900 - 0.433i)T \)
89 \( 1 + (0.900 - 0.433i)T \)
97 \( 1 + iT \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−18.15989421262820697588229564640, −17.6208354422352312501938675903, −16.689200204787037797998925819356, −16.41806712725579839799837581759, −15.08431189690480496076554232040, −14.60796686626585819011991457339, −14.10448512137720715658388269586, −13.34279605713307672596711005615, −12.357244934866139891580338716804, −12.19870293090193089278117536720, −11.5423652907449193006729900983, −10.40818383828488536539090036118, −9.70324837938287355949401891863, −9.10552033111543727672517135198, −8.106517539467467030085541842252, −7.60687840298099200534398818125, −6.91666063033767431220194263349, −6.28266271419575351194203497452, −5.40727389536507270543534675228, −4.58091538591139763553730934143, −3.65152975426533828802982158539, −2.76240788665227455018159042396, −1.9600921123840943431243066015, −1.354563124029321202797417359654, −0.23181357098742737875832474365, 0.66963654628212465497951442886, 1.91984782114920167147225805281, 2.880554484175290435644372923218, 3.47522838368130963530452704162, 4.23620322720501271409733189375, 5.06655596151306367018514721166, 5.71292901595539377662323765344, 6.4309115338137843454863271194, 7.61544534440962584866721928008, 8.1050229496513812940332114483, 9.02357298263318395459423899952, 9.48532784326943292820907796588, 10.35914493314783313381313833768, 10.75962597196484142251455808461, 11.7736026021827674965516576155, 12.12631977382227743311682953974, 13.367190818305863568612056451634, 13.85470609759867783805155209708, 14.559842352994254210316893499030, 15.31054244697941486161890805865, 15.68485105892122651225430777052, 16.59838149552581539057607192024, 17.18150914861726461771029768071, 17.55682802453144039792116491559, 18.86029606651020217115934316899

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