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Properties

Label 1-1960-1960.163-r0-0-0
Degree $1$
Conductor $1960$
Sign $-0.831 - 0.556i$
Analytic cond. $9.10220$
Root an. cond. $9.10220$
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  + (−0.149 − 0.988i)3-s + (−0.955 + 0.294i)9-s + (0.955 + 0.294i)11-s + (−0.974 + 0.222i)13-s + (−0.563 − 0.826i)17-s + (0.5 + 0.866i)19-s + (0.563 − 0.826i)23-s + (0.433 + 0.900i)27-s + (−0.900 − 0.433i)29-s + (0.5 − 0.866i)31-s + (0.149 − 0.988i)33-s + (−0.997 + 0.0747i)37-s + (0.365 + 0.930i)39-s + (0.623 + 0.781i)41-s + (0.781 + 0.623i)43-s + ⋯
L(s)  = 1  + (−0.149 − 0.988i)3-s + (−0.955 + 0.294i)9-s + (0.955 + 0.294i)11-s + (−0.974 + 0.222i)13-s + (−0.563 − 0.826i)17-s + (0.5 + 0.866i)19-s + (0.563 − 0.826i)23-s + (0.433 + 0.900i)27-s + (−0.900 − 0.433i)29-s + (0.5 − 0.866i)31-s + (0.149 − 0.988i)33-s + (−0.997 + 0.0747i)37-s + (0.365 + 0.930i)39-s + (0.623 + 0.781i)41-s + (0.781 + 0.623i)43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.831 - 0.556i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.831 - 0.556i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1960\)    =    \(2^{3} \cdot 5 \cdot 7^{2}\)
Sign: $-0.831 - 0.556i$
Analytic conductor: \(9.10220\)
Root analytic conductor: \(9.10220\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1960} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1960,\ (0:\ ),\ -0.831 - 0.556i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2734920007 - 0.9003735408i\)
\(L(\frac12)\) \(\approx\) \(0.2734920007 - 0.9003735408i\)
\(L(1)\) \(\approx\) \(0.8197972285 - 0.3595405642i\)
\(L(1)\) \(\approx\) \(0.8197972285 - 0.3595405642i\)

Euler product

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

−20.18059318993495352253401542395, −19.683681719198443999490878328319, −19.108316688446407834879362683596, −17.74477772286375862918534191444, −17.30589092130786845754285124437, −16.777534250393097858574128355536, −15.71021107283113045070428126526, −15.372126876119512527327752701407, −14.43956286595548321327236998348, −13.96580833747948060211692609510, −12.83107376983656633079950541930, −12.05835667251709217907593759831, −11.22605220182440485501264582323, −10.73207185909388494122120395944, −9.75023367629515110169750630029, −9.16466377839948838190775334786, −8.56203671976549743335551914888, −7.388132260612691769311162765223, −6.61469083502203269803781249350, −5.612615875670818014792553189848, −4.98597813613852438557752654166, −4.08791173770849722209618612926, −3.382078928658827350412904817481, −2.46453558685771519570960172300, −1.16986838554856557824380813315, 0.34902140649029950387241038627, 1.533596749236149255028073618592, 2.286086241905071093066647918184, 3.21002505176831909674063370117, 4.38688155533076055375375010789, 5.18130164948182987151349777375, 6.19866451362383279700920291979, 6.812582685137353184026413192968, 7.51682232096377901602708605956, 8.251948223176006964779294976860, 9.285366598428830977699170282858, 9.79448260976411990169611823033, 11.10345618771063830924337066670, 11.60529367141937536667325369803, 12.3729383733706978895893063482, 12.903053912083512908754236062293, 13.94023155767785472371837641950, 14.35999331605213649541900692156, 15.14474847117338792876074179456, 16.26277666007983393354424361123, 17.007403865523796120336447785852, 17.42933512848806363715724400516, 18.36601432071654617933633593923, 18.90932103085238401512639146871, 19.632530990921878296384189413792

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