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Properties

Label 1-42e2-1764.1003-r1-0-0
Degree $1$
Conductor $1764$
Sign $-0.110 + 0.993i$
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  + (0.826 − 0.563i)5-s + (0.988 + 0.149i)11-s + (−0.988 − 0.149i)13-s + (−0.733 − 0.680i)17-s + (0.5 + 0.866i)19-s + (−0.955 − 0.294i)23-s + (0.365 − 0.930i)25-s + (−0.733 − 0.680i)29-s − 31-s + (0.955 − 0.294i)37-s + (0.0747 + 0.997i)41-s + (−0.0747 + 0.997i)43-s + (−0.623 + 0.781i)47-s + (0.955 + 0.294i)53-s + (0.900 − 0.433i)55-s + ⋯
L(s)  = 1  + (0.826 − 0.563i)5-s + (0.988 + 0.149i)11-s + (−0.988 − 0.149i)13-s + (−0.733 − 0.680i)17-s + (0.5 + 0.866i)19-s + (−0.955 − 0.294i)23-s + (0.365 − 0.930i)25-s + (−0.733 − 0.680i)29-s − 31-s + (0.955 − 0.294i)37-s + (0.0747 + 0.997i)41-s + (−0.0747 + 0.997i)43-s + (−0.623 + 0.781i)47-s + (0.955 + 0.294i)53-s + (0.900 − 0.433i)55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.110 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.110 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.110 + 0.993i$
Analytic conductor: \(189.568\)
Root analytic conductor: \(189.568\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (1003, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1764,\ (1:\ ),\ -0.110 + 0.993i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8245807520 + 0.9210542429i\)
\(L(\frac12)\) \(\approx\) \(0.8245807520 + 0.9210542429i\)
\(L(1)\) \(\approx\) \(1.082581346 + 0.03106472715i\)
\(L(1)\) \(\approx\) \(1.082581346 + 0.03106472715i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (0.826 - 0.563i)T \)
11 \( 1 + (0.988 + 0.149i)T \)
13 \( 1 + (-0.988 - 0.149i)T \)
17 \( 1 + (-0.733 - 0.680i)T \)
19 \( 1 + (0.5 + 0.866i)T \)
23 \( 1 + (-0.955 - 0.294i)T \)
29 \( 1 + (-0.733 - 0.680i)T \)
31 \( 1 - T \)
37 \( 1 + (0.955 - 0.294i)T \)
41 \( 1 + (0.0747 + 0.997i)T \)
43 \( 1 + (-0.0747 + 0.997i)T \)
47 \( 1 + (-0.623 + 0.781i)T \)
53 \( 1 + (0.955 + 0.294i)T \)
59 \( 1 + (0.900 - 0.433i)T \)
61 \( 1 + (-0.222 + 0.974i)T \)
67 \( 1 - T \)
71 \( 1 + (0.222 + 0.974i)T \)
73 \( 1 + (-0.988 + 0.149i)T \)
79 \( 1 - T \)
83 \( 1 + (0.988 - 0.149i)T \)
89 \( 1 + (0.365 - 0.930i)T \)
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

−19.80464520792068704182108885652, −19.23177336150620755632375337052, −18.20726004526695172280298125546, −17.711696239857111654138388042555, −16.9882227618593090409920882697, −16.35335820532050795085998642872, −15.129028617787050990815520835739, −14.74331833578006501214279637738, −13.89816032297203620457745967269, −13.31102238831413405746225385401, −12.37698012519522241132963774223, −11.54182898346685861362704873159, −10.83559401384872218069360269611, −9.96588826969819379187316897261, −9.31459953840648705475195838455, −8.6573590867778467219786195204, −7.336229602919190612604901182420, −6.88273970068495300746687292858, −5.97638476009744567706736587618, −5.25373077345672779198384883374, −4.17524233630884787024284204988, −3.32091375299577650632827555771, −2.233338480623196146441090618267, −1.650564676077797808188897266554, −0.21685068243491896309279339776, 1.01020289290334784822875733234, 1.91799494657134245878955236006, 2.71289950233348411086417244311, 4.01229252715079911928121204082, 4.68075058687163186280878092997, 5.6527944550004907885085385562, 6.27216810741187350004080192377, 7.2550511146255282711029191362, 8.065793957729111423576276080664, 9.132061993030065831609262360907, 9.58185962927631848789389747096, 10.22469305203516608263358565019, 11.43196462606792920258682248556, 12.0084232386431138138197604823, 12.87143622568947573687813794277, 13.48342256195995601534024058984, 14.52695615028932648864563244563, 14.69350068357662980059202239484, 16.161747362680878460582160946402, 16.483961621691300691474441337046, 17.38368958063050102604409277263, 17.90817219668800610160519606504, 18.672274037313442363513682125420, 19.87464855776225910948505189963, 20.06834469236347683508259084633

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