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Properties

Label 1-1380-1380.287-r0-0-0
Degree $1$
Conductor $1380$
Sign $0.371 + 0.928i$
Analytic cond. $6.40869$
Root an. cond. $6.40869$
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  + (−0.989 − 0.142i)7-s + (−0.415 − 0.909i)11-s + (−0.989 + 0.142i)13-s + (−0.755 − 0.654i)17-s + (0.654 + 0.755i)19-s + (−0.654 + 0.755i)29-s + (0.959 − 0.281i)31-s + (0.540 − 0.841i)37-s + (−0.841 + 0.540i)41-s + (−0.281 + 0.959i)43-s − i·47-s + (0.959 + 0.281i)49-s + (0.989 + 0.142i)53-s + (0.142 + 0.989i)59-s + (0.959 − 0.281i)61-s + ⋯
L(s)  = 1  + (−0.989 − 0.142i)7-s + (−0.415 − 0.909i)11-s + (−0.989 + 0.142i)13-s + (−0.755 − 0.654i)17-s + (0.654 + 0.755i)19-s + (−0.654 + 0.755i)29-s + (0.959 − 0.281i)31-s + (0.540 − 0.841i)37-s + (−0.841 + 0.540i)41-s + (−0.281 + 0.959i)43-s − i·47-s + (0.959 + 0.281i)49-s + (0.989 + 0.142i)53-s + (0.142 + 0.989i)59-s + (0.959 − 0.281i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.371 + 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.371 + 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1380\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.371 + 0.928i$
Analytic conductor: \(6.40869\)
Root analytic conductor: \(6.40869\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1380} (287, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1380,\ (0:\ ),\ 0.371 + 0.928i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6493782743 + 0.4396631261i\)
\(L(\frac12)\) \(\approx\) \(0.6493782743 + 0.4396631261i\)
\(L(1)\) \(\approx\) \(0.8055138895 + 0.02711513857i\)
\(L(1)\) \(\approx\) \(0.8055138895 + 0.02711513857i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
23 \( 1 \)
good7 \( 1 + (-0.989 - 0.142i)T \)
11 \( 1 + (-0.415 - 0.909i)T \)
13 \( 1 + (-0.989 + 0.142i)T \)
17 \( 1 + (-0.755 - 0.654i)T \)
19 \( 1 + (0.654 + 0.755i)T \)
29 \( 1 + (-0.654 + 0.755i)T \)
31 \( 1 + (0.959 - 0.281i)T \)
37 \( 1 + (0.540 - 0.841i)T \)
41 \( 1 + (-0.841 + 0.540i)T \)
43 \( 1 + (-0.281 + 0.959i)T \)
47 \( 1 - iT \)
53 \( 1 + (0.989 + 0.142i)T \)
59 \( 1 + (0.142 + 0.989i)T \)
61 \( 1 + (0.959 - 0.281i)T \)
67 \( 1 + (0.909 + 0.415i)T \)
71 \( 1 + (0.415 - 0.909i)T \)
73 \( 1 + (-0.755 + 0.654i)T \)
79 \( 1 + (0.142 + 0.989i)T \)
83 \( 1 + (-0.540 + 0.841i)T \)
89 \( 1 + (0.959 + 0.281i)T \)
97 \( 1 + (-0.540 - 0.841i)T \)
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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

−20.4259477877747731388959415049, −20.01914617916494004314431491716, −19.20876567077268292680427720512, −18.52116048830302316961985270537, −17.485555451668442255421414940211, −17.10501089267049462885808505029, −15.998468082124136508309147411362, −15.35013071219375364820067001241, −14.84696227485175576329138566573, −13.541373016747948759267155962172, −13.12682193717731706338704189644, −12.23485145151303849779347073564, −11.61620780265133163745051835664, −10.35005385638127840190027349441, −9.91386543010226261443880136371, −9.11616777438388302775545367225, −8.15065705714296332794738422061, −7.12332590487785743115049883407, −6.656574507709797589465832590200, −5.51285483739948828939534815329, −4.74114547980879607635375490482, −3.75244211098302093636075977134, −2.70561602419753406773346145102, −2.01797428556976450631530126398, −0.36078551206216405107277204235, 0.918889166090316960085690203106, 2.41775534526066215502217196087, 3.08996678456943018522495350625, 4.04649418552835452683818355944, 5.10958927664465011350712307121, 5.923861752225869201897951737633, 6.804637975048728486028638355371, 7.55200591789504149195899503355, 8.50111021760059364049233825018, 9.472802544099444440606287217394, 9.96132953238678805596216475315, 10.9592080298785196886968944479, 11.73792820897099190169906314157, 12.61710040435041087083076698519, 13.32583913830937205476839432831, 14.01604216782933980135606748505, 14.88342523306571284094996186089, 15.86881781318447653362060082110, 16.35609799224144666574136434681, 17.03759731338878628553012997637, 18.10194153365042132799857893999, 18.71963561449335711042149668963, 19.53129236229059493340963108490, 20.07118913523137387229563131864, 20.988261133490212009494772090648

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