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Properties

Label 1-42e2-1764.1427-r1-0-0
Degree $1$
Conductor $1764$
Sign $-0.645 - 0.763i$
Analytic cond. $189.568$
Root an. cond. $189.568$
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  + (0.365 − 0.930i)5-s + (0.955 + 0.294i)11-s + (0.733 − 0.680i)13-s + (−0.900 − 0.433i)17-s + 19-s + (0.826 + 0.563i)23-s + (−0.733 − 0.680i)25-s + (−0.826 + 0.563i)29-s + (−0.5 − 0.866i)31-s + (−0.900 − 0.433i)37-s + (0.365 − 0.930i)41-s + (−0.365 − 0.930i)43-s + (−0.955 − 0.294i)47-s + (0.900 − 0.433i)53-s + (0.623 − 0.781i)55-s + ⋯
L(s)  = 1  + (0.365 − 0.930i)5-s + (0.955 + 0.294i)11-s + (0.733 − 0.680i)13-s + (−0.900 − 0.433i)17-s + 19-s + (0.826 + 0.563i)23-s + (−0.733 − 0.680i)25-s + (−0.826 + 0.563i)29-s + (−0.5 − 0.866i)31-s + (−0.900 − 0.433i)37-s + (0.365 − 0.930i)41-s + (−0.365 − 0.930i)43-s + (−0.955 − 0.294i)47-s + (0.900 − 0.433i)53-s + (0.623 − 0.781i)55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.645 - 0.763i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.645 - 0.763i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.645 - 0.763i$
Analytic conductor: \(189.568\)
Root analytic conductor: \(189.568\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (1427, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1764,\ (1:\ ),\ -0.645 - 0.763i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8773422845 - 1.890248453i\)
\(L(\frac12)\) \(\approx\) \(0.8773422845 - 1.890248453i\)
\(L(1)\) \(\approx\) \(1.134638482 - 0.3742382311i\)
\(L(1)\) \(\approx\) \(1.134638482 - 0.3742382311i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (0.365 - 0.930i)T \)
11 \( 1 + (0.955 + 0.294i)T \)
13 \( 1 + (0.733 - 0.680i)T \)
17 \( 1 + (-0.900 - 0.433i)T \)
19 \( 1 + T \)
23 \( 1 + (0.826 + 0.563i)T \)
29 \( 1 + (-0.826 + 0.563i)T \)
31 \( 1 + (-0.5 - 0.866i)T \)
37 \( 1 + (-0.900 - 0.433i)T \)
41 \( 1 + (0.365 - 0.930i)T \)
43 \( 1 + (-0.365 - 0.930i)T \)
47 \( 1 + (-0.955 - 0.294i)T \)
53 \( 1 + (0.900 - 0.433i)T \)
59 \( 1 + (0.988 + 0.149i)T \)
61 \( 1 + (-0.826 + 0.563i)T \)
67 \( 1 + (0.5 + 0.866i)T \)
71 \( 1 + (-0.900 + 0.433i)T \)
73 \( 1 + (0.222 - 0.974i)T \)
79 \( 1 + (0.5 - 0.866i)T \)
83 \( 1 + (0.733 + 0.680i)T \)
89 \( 1 + (-0.222 + 0.974i)T \)
97 \( 1 + (0.5 - 0.866i)T \)
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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

−20.24593851274024651100802423533, −19.5186891501301929514065616340, −18.79173210108399583991649932592, −18.18317842593353613421903829918, −17.453228959380287402909068596656, −16.66960771021148933022171391195, −15.910269607255502696801822064306, −14.97968578583924667002871006614, −14.42327127746212675879304898632, −13.68071243319566118441936874185, −13.07603124808392593231695579067, −11.886222677713575619307270635421, −11.23332939560924148626484320134, −10.72896349308346860163777025892, −9.6421867965645508045844522093, −9.0862670508361426151914551258, −8.19728494269322979594613356465, −7.04566049484539584891549757939, −6.57329102689295755102686076148, −5.8724167047694164777461809509, −4.75093836884477773805870541820, −3.74012372651122536578707549606, −3.10451919179547526800606240882, −1.972975788491631797227614281324, −1.17604526213867022078368101223, 0.38196569056468746158031812668, 1.28392254843370339033278154010, 2.095645148671942884870827455711, 3.40872609516645495693517128747, 4.118046300291142765676306338677, 5.21772229666113136633977073497, 5.64754062602935380014485629154, 6.78288381466491955817026044591, 7.49280776815673530699689129955, 8.6410526000340167093278463649, 9.08701053212725740912902576443, 9.77773691668216374410664776423, 10.82169523327580862214511849244, 11.61254173314532819205696368282, 12.30723724911896855590255681170, 13.29063482478288804348761097449, 13.53666473503203917510600159207, 14.628736506675881729585149825780, 15.42385411535772313003779394984, 16.17245485352266348018028750245, 16.830910185028222899601875833038, 17.65430771320153900648525541826, 18.07059439296332114029100526747, 19.17640901947632753722641941379, 19.96227838352202468485416962221

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