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Properties

Label 1-1380-1380.1103-r0-0-0
Degree $1$
Conductor $1380$
Sign $-0.850 + 0.525i$
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  − i·7-s − 11-s + i·13-s − i·17-s − 19-s + 29-s − 31-s + i·37-s − 41-s + i·43-s − i·47-s − 49-s + i·53-s − 59-s − 61-s + ⋯
L(s)  = 1  − i·7-s − 11-s + i·13-s − i·17-s − 19-s + 29-s − 31-s + i·37-s − 41-s + i·43-s − i·47-s − 49-s + i·53-s − 59-s − 61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.850 + 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.850 + 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1380\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.850 + 0.525i$
Analytic conductor: \(6.40869\)
Root analytic conductor: \(6.40869\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1380} (1103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1380,\ (0:\ ),\ -0.850 + 0.525i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.03830942612 + 0.1348548122i\)
\(L(\frac12)\) \(\approx\) \(0.03830942612 + 0.1348548122i\)
\(L(1)\) \(\approx\) \(0.7774104452 - 0.04766107499i\)
\(L(1)\) \(\approx\) \(0.7774104452 - 0.04766107499i\)

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 \( 1 \)
11 \( 1 \)
13 \( 1 \)
17 \( 1 \)
19 \( 1 - iT \)
29 \( 1 \)
31 \( 1 \)
37 \( 1 - T \)
41 \( 1 \)
43 \( 1 + iT \)
47 \( 1 \)
53 \( 1 \)
59 \( 1 \)
61 \( 1 - iT \)
67 \( 1 \)
71 \( 1 - T \)
73 \( 1 \)
79 \( 1 \)
83 \( 1 \)
89 \( 1 \)
97 \( 1 \)
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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

−20.61286039109536322386410294834, −19.67459363036656204933667802608, −19.011384811773458261458762499519, −18.204923643085015242396659651358, −17.66326753089276495588779615815, −16.74799238953460094468991785803, −15.75150238101366737400486316193, −15.2822323608318184631329720406, −14.62238136196256665703681417144, −13.52180882153785707049459784947, −12.55969581401634599879190382245, −12.46499454858465312006155555474, −11.07184920869802278652471728960, −10.550667747473064253013173013653, −9.685029452868573578915103479762, −8.54075551626280376862520854304, −8.24122467951629954526740724211, −7.15991022801519378061559110904, −6.03274791258236621019932896679, −5.525026098804263756121073228254, −4.59775599092969578760637210075, −3.42226315485556351308137053767, −2.5729830509538867328661187038, −1.72848120574002898056173754312, −0.051578988228225601488025118008, 1.33601727162627164439862532628, 2.426777381295018796154762611130, 3.40283521274873333351786387928, 4.463298218534429423599834624482, 4.9898980045449165042187811913, 6.278544611587921432574273172634, 6.989535753402340651471841527788, 7.75071288149972188552777417448, 8.62054769948759405258208312736, 9.580867848056517336607206948261, 10.36417544731906270668041037469, 11.02027434352566311931554905358, 11.86962252508924813385728195154, 12.81991828293232312256076882729, 13.60663950939625713278389363213, 14.096114113827146544823906101381, 15.081622016732097829893009556556, 15.92672841840245060425755634525, 16.658922077576046667043648264876, 17.20619497834119359603298252303, 18.30392359723409249586705022388, 18.71675935230276150404246979395, 19.80727924228498933544690457332, 20.29733486162613454231012028605, 21.224430573337463098370045042859

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