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Properties

Label 1-1580-1580.783-r0-0-0
Degree $1$
Conductor $1580$
Sign $-0.962 - 0.269i$
Analytic cond. $7.33748$
Root an. cond. $7.33748$
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  + (0.903 − 0.428i)3-s + (0.0804 − 0.996i)7-s + (0.632 − 0.774i)9-s + (−0.278 − 0.960i)11-s + (−0.600 + 0.799i)13-s + (−0.992 − 0.120i)17-s + (−0.0402 − 0.999i)19-s + (−0.354 − 0.935i)21-s + (−0.866 + 0.5i)23-s + (0.239 − 0.970i)27-s + (−0.987 + 0.160i)29-s + (−0.948 − 0.316i)31-s + (−0.663 − 0.748i)33-s + (0.534 + 0.845i)37-s + (−0.200 + 0.979i)39-s + ⋯
L(s)  = 1  + (0.903 − 0.428i)3-s + (0.0804 − 0.996i)7-s + (0.632 − 0.774i)9-s + (−0.278 − 0.960i)11-s + (−0.600 + 0.799i)13-s + (−0.992 − 0.120i)17-s + (−0.0402 − 0.999i)19-s + (−0.354 − 0.935i)21-s + (−0.866 + 0.5i)23-s + (0.239 − 0.970i)27-s + (−0.987 + 0.160i)29-s + (−0.948 − 0.316i)31-s + (−0.663 − 0.748i)33-s + (0.534 + 0.845i)37-s + (−0.200 + 0.979i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1580 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.962 - 0.269i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1580 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.962 - 0.269i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1580\)    =    \(2^{2} \cdot 5 \cdot 79\)
Sign: $-0.962 - 0.269i$
Analytic conductor: \(7.33748\)
Root analytic conductor: \(7.33748\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1580} (783, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1580,\ (0:\ ),\ -0.962 - 0.269i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1613443017 - 1.173678817i\)
\(L(\frac12)\) \(\approx\) \(0.1613443017 - 1.173678817i\)
\(L(1)\) \(\approx\) \(1.032616690 - 0.4918938047i\)
\(L(1)\) \(\approx\) \(1.032616690 - 0.4918938047i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
79 \( 1 \)
good3 \( 1 + (0.903 - 0.428i)T \)
7 \( 1 + (0.0804 - 0.996i)T \)
11 \( 1 + (-0.278 - 0.960i)T \)
13 \( 1 + (-0.600 + 0.799i)T \)
17 \( 1 + (-0.992 - 0.120i)T \)
19 \( 1 + (-0.0402 - 0.999i)T \)
23 \( 1 + (-0.866 + 0.5i)T \)
29 \( 1 + (-0.987 + 0.160i)T \)
31 \( 1 + (-0.948 - 0.316i)T \)
37 \( 1 + (0.534 + 0.845i)T \)
41 \( 1 + (-0.970 + 0.239i)T \)
43 \( 1 + (-0.960 - 0.278i)T \)
47 \( 1 + (0.534 - 0.845i)T \)
53 \( 1 + (0.903 + 0.428i)T \)
59 \( 1 + (-0.919 - 0.391i)T \)
61 \( 1 + (0.885 + 0.464i)T \)
67 \( 1 + (0.663 - 0.748i)T \)
71 \( 1 + (-0.568 - 0.822i)T \)
73 \( 1 + (0.600 + 0.799i)T \)
83 \( 1 + (0.391 + 0.919i)T \)
89 \( 1 + (-0.568 + 0.822i)T \)
97 \( 1 + (0.464 - 0.885i)T \)
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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

−20.63532200952576821167142405012, −20.212708108115672247502511803523, −19.530449931637319856377582732638, −18.510673914434015502527445884766, −18.12793243563039556311588392930, −17.11037381429551280660427751997, −16.094878718828097844516674798747, −15.49356004740046195229695208799, −14.79634904704039358026337423331, −14.439579497375486034720785636175, −13.16436856610022170096606139023, −12.69946950881213120222139094362, −11.88324285320509084305685408524, −10.7446515790866000486090777187, −10.023015030772054869388188833216, −9.368154255022310020206527322451, −8.55476198928588465095420494480, −7.8734878688106723799343721006, −7.107956364750951328849159613380, −5.876645925138952051198939473824, −5.08135051056726278392830154921, −4.24193315791248062917835047921, −3.3211583473644234106433577894, −2.24284706694850636512140798584, −1.93378697571845162179200784796, 0.338491346051369883325538195983, 1.62294167778276831398241798249, 2.42737569441162584788143878387, 3.48558371906243597488647502085, 4.13060942725388385528977817018, 5.12415852639584089798782559506, 6.43554492271195868971893943190, 7.04817805876386707493132270881, 7.7504845205136631038429762213, 8.60964367779756910470493564054, 9.31105821824099108782933542155, 10.10899721088861938330914646540, 11.11877870352394029250712688235, 11.73315278260830840819512373232, 12.92214166923316491875477661851, 13.58793591565613333141536435621, 13.85619455211993175416063754923, 14.86306086469946185219966380770, 15.51575736484631793458909344414, 16.5342668552816786721139043496, 17.10306914372084986752141437380, 18.16657594365614457385143971624, 18.655289820578984223353341960228, 19.723918105385084273794119957657, 19.92003511112116613371425348967

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