Properties

Label 2-42e2-21.5-c3-0-11
Degree $2$
Conductor $1764$
Sign $0.778 - 0.627i$
Analytic cond. $104.079$
Root an. cond. $10.2019$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (7.66 − 13.2i)5-s + (−17.6 + 10.1i)11-s + 30.5i·13-s + (−1.72 − 2.98i)17-s + (127. + 73.6i)19-s + (110. + 63.5i)23-s + (−55.0 − 95.3i)25-s − 216. i·29-s + (−89.6 + 51.7i)31-s + (−118. + 204. i)37-s − 480.·41-s + 147.·43-s + (−227. + 394. i)47-s + (−173. + 100. i)53-s + 311. i·55-s + ⋯
L(s)  = 1  + (0.685 − 1.18i)5-s + (−0.482 + 0.278i)11-s + 0.651i·13-s + (−0.0246 − 0.0426i)17-s + (1.54 + 0.889i)19-s + (0.998 + 0.576i)23-s + (−0.440 − 0.762i)25-s − 1.38i·29-s + (−0.519 + 0.299i)31-s + (−0.525 + 0.910i)37-s − 1.82·41-s + 0.524·43-s + (−0.706 + 1.22i)47-s + (−0.450 + 0.260i)53-s + 0.764i·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.778 - 0.627i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.778 - 0.627i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.778 - 0.627i$
Analytic conductor: \(104.079\)
Root analytic conductor: \(10.2019\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (1097, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1764,\ (\ :3/2),\ 0.778 - 0.627i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.175922276\)
\(L(\frac12)\) \(\approx\) \(2.175922276\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (-7.66 + 13.2i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (17.6 - 10.1i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 - 30.5iT - 2.19e3T^{2} \)
17 \( 1 + (1.72 + 2.98i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-127. - 73.6i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-110. - 63.5i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + 216. iT - 2.43e4T^{2} \)
31 \( 1 + (89.6 - 51.7i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (118. - 204. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + 480.T + 6.89e4T^{2} \)
43 \( 1 - 147.T + 7.95e4T^{2} \)
47 \( 1 + (227. - 394. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (173. - 100. i)T + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-80.1 - 138. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-712. - 411. i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-203. - 352. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 513. iT - 3.57e5T^{2} \)
73 \( 1 + (410. - 236. i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-60.6 + 105. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 - 536.T + 5.71e5T^{2} \)
89 \( 1 + (-115. + 200. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 - 734. iT - 9.12e5T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.122196874233232235273371153357, −8.311057178693339121396504833478, −7.52873609072488793389877261120, −6.60817018598370096614330864363, −5.48993479832363615574790745743, −5.16731226877203910029167899742, −4.16471185034590982752856390046, −3.01730178265464556857235678792, −1.75868062492729732861754804425, −1.04258562139298334236928713965, 0.49788572916356760919049940646, 1.91335940543023349479223906589, 2.98557615958298037579743609617, 3.40170912048320334014999271409, 5.09657514559525767883209206092, 5.46237258027670688985552747752, 6.70738669848177907969607544926, 7.02028424038081048891936944224, 7.993074375092428267400150711398, 8.950100603574426454710382279135

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