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Properties

Label 1-1380-1380.803-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: $\chi_{1380} (803, \cdot )$
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

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

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