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Properties

Label 1-3920-3920.3-r1-0-0
Degree $1$
Conductor $3920$
Sign $-0.849 + 0.528i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.849 + 0.528i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.849 + 0.528i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3920\)    =    \(2^{4} \cdot 5 \cdot 7^{2}\)
Sign: $-0.849 + 0.528i$
Analytic conductor: \(421.262\)
Root analytic conductor: \(421.262\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3920} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3920,\ (1:\ ),\ -0.849 + 0.528i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8672235054 + 3.037314721i\)
\(L(\frac12)\) \(\approx\) \(0.8672235054 + 3.037314721i\)
\(L(1)\) \(\approx\) \(1.423303406 + 0.5589235970i\)
\(L(1)\) \(\approx\) \(1.423303406 + 0.5589235970i\)

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

−18.09380569559810111897321824542, −17.65692168737561373598843629357, −16.494102189427113165788926964126, −15.98102489962460004816595876886, −15.330397499982797940707251770, −14.49932527678236099693034646877, −14.0972976449937477558472307230, −13.178064695583938187349435261054, −12.94532272719549878234065264482, −11.85173968756925544019056947681, −11.21575078737739616408982398680, −10.33000089329623593902432599135, −9.63710525971212382702595117102, −8.89713505511473815504036726229, −8.38949949742394602608828478343, −7.59627757321687844303331387065, −6.94979256894220596124235710275, −6.15773242233333611958454805589, −5.150663830268385146284761919497, −4.47213764489667301837309381763, −3.36226787417940818742258755079, −2.99491512851820466377969792612, −2.18186674559210692505483536424, −1.037758839142139771645162600205, −0.391896908742333168927240608171, 1.34847782417357287484429993307, 1.789625120947426600462542727070, 2.7112326555743961008617490674, 3.585567442283975892189135378427, 4.252694171058894504226668790975, 4.89330944966804986507696855462, 5.94009055215038997093860739979, 7.03684378619151270897117056799, 7.26860896443719293206559307230, 8.24289285280398863823407380587, 8.96896370990356027012957784934, 9.52956716876734572282333977675, 10.11090406227448685295038160954, 10.979766311885515727362801094945, 11.86306193988840522294787109948, 12.525975333923650717418960222381, 13.27038704999847628016757942534, 13.92323812544381846776934633037, 14.580246702873441811505799935552, 15.13820234187984398520840424040, 15.74297366513460697064716897423, 16.565782495380091633515454034665, 17.2397783676656945181310324563, 18.06273102151629607431335967239, 18.73185085811093232394436467083

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