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Properties

Label 1-3920-3920.2613-r1-0-0
Degree $1$
Conductor $3920$
Sign $0.852 - 0.523i$
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  + (−0.988 + 0.149i)3-s + (0.955 − 0.294i)9-s + (−0.294 + 0.955i)11-s + (−0.222 − 0.974i)13-s + (−0.563 − 0.826i)17-s + (0.866 − 0.5i)19-s + (0.563 − 0.826i)23-s + (−0.900 + 0.433i)27-s + (0.433 − 0.900i)29-s + (−0.5 + 0.866i)31-s + (0.149 − 0.988i)33-s + (0.0747 + 0.997i)37-s + (0.365 + 0.930i)39-s + (−0.623 − 0.781i)41-s + (−0.623 + 0.781i)43-s + ⋯
L(s)  = 1  + (−0.988 + 0.149i)3-s + (0.955 − 0.294i)9-s + (−0.294 + 0.955i)11-s + (−0.222 − 0.974i)13-s + (−0.563 − 0.826i)17-s + (0.866 − 0.5i)19-s + (0.563 − 0.826i)23-s + (−0.900 + 0.433i)27-s + (0.433 − 0.900i)29-s + (−0.5 + 0.866i)31-s + (0.149 − 0.988i)33-s + (0.0747 + 0.997i)37-s + (0.365 + 0.930i)39-s + (−0.623 − 0.781i)41-s + (−0.623 + 0.781i)43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.852 - 0.523i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.852 - 0.523i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3920\)    =    \(2^{4} \cdot 5 \cdot 7^{2}\)
Sign: $0.852 - 0.523i$
Analytic conductor: \(421.262\)
Root analytic conductor: \(421.262\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3920} (2613, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3920,\ (1:\ ),\ 0.852 - 0.523i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.265347568 - 0.3576328846i\)
\(L(\frac12)\) \(\approx\) \(1.265347568 - 0.3576328846i\)
\(L(1)\) \(\approx\) \(0.7823631497 + 0.01796859233i\)
\(L(1)\) \(\approx\) \(0.7823631497 + 0.01796859233i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good3 \( 1 + (-0.988 + 0.149i)T \)
11 \( 1 + (-0.294 + 0.955i)T \)
13 \( 1 + (-0.222 - 0.974i)T \)
17 \( 1 + (-0.563 - 0.826i)T \)
19 \( 1 + (0.866 - 0.5i)T \)
23 \( 1 + (0.563 - 0.826i)T \)
29 \( 1 + (0.433 - 0.900i)T \)
31 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 + (0.0747 + 0.997i)T \)
41 \( 1 + (-0.623 - 0.781i)T \)
43 \( 1 + (-0.623 + 0.781i)T \)
47 \( 1 + (0.680 + 0.733i)T \)
53 \( 1 + (-0.0747 + 0.997i)T \)
59 \( 1 + (0.930 - 0.365i)T \)
61 \( 1 + (0.997 - 0.0747i)T \)
67 \( 1 + (0.5 - 0.866i)T \)
71 \( 1 + (0.900 - 0.433i)T \)
73 \( 1 + (-0.680 + 0.733i)T \)
79 \( 1 + (0.5 + 0.866i)T \)
83 \( 1 + (-0.222 + 0.974i)T \)
89 \( 1 + (0.955 - 0.294i)T \)
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

−18.4510731219579838854768467324, −17.65589301952903397268284038071, −17.06798661773772143881939109239, −16.29111781625882447595333522873, −16.05479179578569636163473074592, −15.04764913114530497707048000992, −14.315257625755605641639412106051, −13.3397311051495736742326438279, −13.08929818576634033945567962733, −11.97246056916729576304615852789, −11.60871705809928046186942714892, −10.89735725909208651325526387741, −10.2758596188797972138333403559, −9.43556305506118996396413756573, −8.67759963926892993030547283029, −7.79915041381087037428461282564, −7.00360722017518207803348203610, −6.44664410670997298261609422711, −5.522751752464395427492823857976, −5.18370927504989633980613788939, −4.065917854637591718233379740383, −3.49889267848497169564263623139, −2.22588582363118332771870117125, −1.453251227969327007278332599929, −0.54872533377662130221972867260, 0.4152519453710168907995296291, 1.131466271234294104647803554178, 2.31617014451578216026294550606, 3.08128324994324824319792404494, 4.19379429517780451270384921508, 5.02141461757871462830804068359, 5.21397741109765468927760142866, 6.35184449682332247801647855623, 6.98419595809096277948018906802, 7.56139421398878616989528839649, 8.519477353288435619747712248731, 9.54755942037111532159445763730, 9.98230137243192552171309803647, 10.73829214971527925842173460616, 11.3714959131347053298406500535, 12.13481197976320757870409087914, 12.67127682699382779920360321821, 13.33407603564971061690025024054, 14.1813898959129377417850758080, 15.27075010730656684661858823608, 15.493101233029905954495840144629, 16.232053451155167786678195830325, 17.08470305695093404397370153273, 17.60474636597411105789224407291, 18.15227149949558446510625268728

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