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Properties

Label 1-1380-1380.83-r0-0-0
Degree $1$
Conductor $1380$
Sign $0.909 - 0.415i$
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  + (−0.755 + 0.654i)7-s + (−0.841 − 0.540i)11-s + (−0.755 − 0.654i)13-s + (0.909 − 0.415i)17-s + (−0.415 + 0.909i)19-s + (0.415 + 0.909i)29-s + (0.142 + 0.989i)31-s + (0.281 − 0.959i)37-s + (0.959 − 0.281i)41-s + (−0.989 − 0.142i)43-s − i·47-s + (0.142 − 0.989i)49-s + (0.755 − 0.654i)53-s + (0.654 − 0.755i)59-s + (0.142 + 0.989i)61-s + ⋯
L(s)  = 1  + (−0.755 + 0.654i)7-s + (−0.841 − 0.540i)11-s + (−0.755 − 0.654i)13-s + (0.909 − 0.415i)17-s + (−0.415 + 0.909i)19-s + (0.415 + 0.909i)29-s + (0.142 + 0.989i)31-s + (0.281 − 0.959i)37-s + (0.959 − 0.281i)41-s + (−0.989 − 0.142i)43-s − i·47-s + (0.142 − 0.989i)49-s + (0.755 − 0.654i)53-s + (0.654 − 0.755i)59-s + (0.142 + 0.989i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.909 - 0.415i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.909 - 0.415i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1380\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.909 - 0.415i$
Analytic conductor: \(6.40869\)
Root analytic conductor: \(6.40869\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1380} (83, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1380,\ (0:\ ),\ 0.909 - 0.415i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.075065837 - 0.2337475622i\)
\(L(\frac12)\) \(\approx\) \(1.075065837 - 0.2337475622i\)
\(L(1)\) \(\approx\) \(0.9034069254 + 0.01867460801i\)
\(L(1)\) \(\approx\) \(0.9034069254 + 0.01867460801i\)

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 \( 1 + (-0.755 + 0.654i)T \)
11 \( 1 + (-0.841 - 0.540i)T \)
13 \( 1 + (-0.755 - 0.654i)T \)
17 \( 1 + (0.909 - 0.415i)T \)
19 \( 1 + (-0.415 + 0.909i)T \)
29 \( 1 + (0.415 + 0.909i)T \)
31 \( 1 + (0.142 + 0.989i)T \)
37 \( 1 + (0.281 - 0.959i)T \)
41 \( 1 + (0.959 - 0.281i)T \)
43 \( 1 + (-0.989 - 0.142i)T \)
47 \( 1 - iT \)
53 \( 1 + (0.755 - 0.654i)T \)
59 \( 1 + (0.654 - 0.755i)T \)
61 \( 1 + (0.142 + 0.989i)T \)
67 \( 1 + (-0.540 - 0.841i)T \)
71 \( 1 + (0.841 - 0.540i)T \)
73 \( 1 + (0.909 + 0.415i)T \)
79 \( 1 + (0.654 - 0.755i)T \)
83 \( 1 + (-0.281 + 0.959i)T \)
89 \( 1 + (0.142 - 0.989i)T \)
97 \( 1 + (-0.281 - 0.959i)T \)
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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

−20.89337036295986445904873789207, −20.077485937534349479518232242395, −19.334570166944986911112570896417, −18.81313157085088800059673693214, −17.76839078726331197638453762910, −17.01187482905463138102026492138, −16.47981984863823198514566709033, −15.51082576736379718601606889841, −14.898352848187235642629023339548, −13.91370506333603762466624977506, −13.19624233850560673715091384344, −12.557710338341264916234772454, −11.68342030746306845135857830807, −10.73275410394112551161198962436, −9.883523433145911351765266114079, −9.52477517194969569637806099667, −8.22791454270088753461217750478, −7.49301189808103601687034559353, −6.74487087339361234068104746209, −5.89349687430230692574053369661, −4.765635842748746409449204028729, −4.13017705132603531091960240485, −2.96631289114295358170137918384, −2.217409329041745455335221007418, −0.80507793815529022993436407484, 0.57963611586177387563572881165, 2.097948357075942663604853931, 2.97916369762066054322383534117, 3.62927237024227632821351513203, 5.12528047486984686453114715451, 5.52728883433129272516543520005, 6.5091222792018950944487509266, 7.475931754634137696744388474808, 8.2658072146291981950338690504, 9.07015944243218259708717122947, 10.09391834208777560518766017556, 10.45735417610794378845669273801, 11.67737236729586446332408855662, 12.49263161177584130277050026355, 12.87814376218267047743223400234, 13.96966153351094217482842019194, 14.70482456548500899003604537850, 15.53101478809792971439322807932, 16.27455685892835928714516078792, 16.77700660534958729987145207495, 18.03716825454260534604826549499, 18.40857130810939593990351389390, 19.35857843362278908243313486644, 19.8203701791041478182784421097, 20.997502326085202975863493959184

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