Properties

Label 1-1380-1380.743-r0-0-0
Degree $1$
Conductor $1380$
Sign $-0.546 - 0.837i$
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  + (−0.540 − 0.841i)7-s + (0.142 − 0.989i)11-s + (−0.540 + 0.841i)13-s + (−0.281 + 0.959i)17-s + (0.959 − 0.281i)19-s + (−0.959 − 0.281i)29-s + (−0.415 − 0.909i)31-s + (0.755 − 0.654i)37-s + (0.654 − 0.755i)41-s + (0.909 + 0.415i)43-s i·47-s + (−0.415 + 0.909i)49-s + (0.540 + 0.841i)53-s + (−0.841 − 0.540i)59-s + (−0.415 − 0.909i)61-s + ⋯
L(s)  = 1  + (−0.540 − 0.841i)7-s + (0.142 − 0.989i)11-s + (−0.540 + 0.841i)13-s + (−0.281 + 0.959i)17-s + (0.959 − 0.281i)19-s + (−0.959 − 0.281i)29-s + (−0.415 − 0.909i)31-s + (0.755 − 0.654i)37-s + (0.654 − 0.755i)41-s + (0.909 + 0.415i)43-s i·47-s + (−0.415 + 0.909i)49-s + (0.540 + 0.841i)53-s + (−0.841 − 0.540i)59-s + (−0.415 − 0.909i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.546 - 0.837i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.546 - 0.837i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1380\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.546 - 0.837i$
Analytic conductor: \(6.40869\)
Root analytic conductor: \(6.40869\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1380} (743, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1380,\ (0:\ ),\ -0.546 - 0.837i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4130541376 - 0.7631294392i\)
\(L(\frac12)\) \(\approx\) \(0.4130541376 - 0.7631294392i\)
\(L(1)\) \(\approx\) \(0.8609699359 - 0.2066662543i\)
\(L(1)\) \(\approx\) \(0.8609699359 - 0.2066662543i\)

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

−21.04429106392629462141305229473, −20.16794697086473170971154149398, −19.79104642251461153387803139276, −18.68375540129966226104103019151, −18.10932415740720489514184118549, −17.47487330415969494984334149016, −16.389184082530437465000035112949, −15.79566598096909647469841968187, −15.01513275581758661429606879124, −14.39760122394989457406050602251, −13.28935772759046577683037048406, −12.59304654160925732551948059999, −12.00243554933808395540091541823, −11.15340749083319281185352381647, −9.99299221641903575047270752405, −9.54163239682982290265739597442, −8.74310773338582615535747682078, −7.57528904197355406878724163858, −7.08015921094345742192927519811, −5.90893677990440392301628463851, −5.257072817968205562416875168598, −4.35139253580872450451191607949, −3.08734081082025611610339064985, −2.53754199257012063540416912589, −1.30032895118670656685935338193, 0.34133916128705142036881980336, 1.56582739122309688421164185246, 2.73117577741654009629566327319, 3.75929356958828907779968956802, 4.30881237800900182406374754583, 5.60723233861545459702461988814, 6.29002161705044814497980753939, 7.250368528973715990646472120630, 7.860322794731426341369772739459, 9.1032171810490438810452532818, 9.522351239118220428993501777089, 10.65138294895393422863402586863, 11.17560372109552953732988708375, 12.10922791553629336624877333705, 13.04820031127909793595634416600, 13.68049648840913128027699034129, 14.35075580955104795362698595675, 15.25560105741428796498849915958, 16.24395495132303903060075463013, 16.71984351548957832286933210173, 17.376435379622927199577811575832, 18.42537372785454574178856271, 19.19560416023041822934846830176, 19.72159881100552552156316521170, 20.49051286700083198519862718165

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