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Properties

Label 2-42e2-21.17-c3-0-16
Degree $2$
Conductor $1764$
Sign $0.879 + 0.475i$
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  + (−3.41 − 5.91i)5-s + (−50.5 − 29.1i)11-s + 38.5i·13-s + (−16.1 + 27.9i)17-s + (−107. + 62.2i)19-s + (−174. + 100. i)23-s + (39.2 − 67.9i)25-s − 104. i·29-s + (240. + 138. i)31-s + (23.8 + 41.2i)37-s + 387.·41-s + 272.·43-s + (81.5 + 141. i)47-s + (−313. − 181. i)53-s + 398. i·55-s + ⋯
L(s)  = 1  + (−0.305 − 0.528i)5-s + (−1.38 − 0.799i)11-s + 0.822i·13-s + (−0.230 + 0.398i)17-s + (−1.30 + 0.751i)19-s + (−1.57 + 0.911i)23-s + (0.313 − 0.543i)25-s − 0.668i·29-s + (1.39 + 0.805i)31-s + (0.105 + 0.183i)37-s + 1.47·41-s + 0.966·43-s + (0.253 + 0.438i)47-s + (−0.813 − 0.469i)53-s + 0.976i·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.879 + 0.475i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.879 + 0.475i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.879 + 0.475i$
Analytic conductor: \(104.079\)
Root analytic conductor: \(10.2019\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (521, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1764,\ (\ :3/2),\ 0.879 + 0.475i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.157273923\)
\(L(\frac12)\) \(\approx\) \(1.157273923\)
\(L(\frac{5}{2})\) 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 \( 1 + (3.41 + 5.91i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (50.5 + 29.1i)T + (665.5 + 1.15e3i)T^{2} \)
13 \( 1 - 38.5iT - 2.19e3T^{2} \)
17 \( 1 + (16.1 - 27.9i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (107. - 62.2i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (174. - 100. i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + 104. iT - 2.43e4T^{2} \)
31 \( 1 + (-240. - 138. i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-23.8 - 41.2i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 - 387.T + 6.89e4T^{2} \)
43 \( 1 - 272.T + 7.95e4T^{2} \)
47 \( 1 + (-81.5 - 141. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (313. + 181. i)T + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (-105. + 183. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-202. + 117. i)T + (1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (262. - 454. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 348. iT - 3.57e5T^{2} \)
73 \( 1 + (465. + 268. i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (362. + 628. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 - 392.T + 5.71e5T^{2} \)
89 \( 1 + (430. + 744. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + 978. iT - 9.12e5T^{2} \)
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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

−8.570046639471749919761244005627, −8.259329839895689707571650077130, −7.50069202246513922441863964119, −6.23785049655381680701459342085, −5.81092514030520082228904596703, −4.58611141444196359345918629027, −4.06084151004276446497652089780, −2.80239958936391754475198939510, −1.81883608636843528159775702868, −0.44104839799206860300686691549, 0.53719540107026925174478833319, 2.32602409398479190411524967591, 2.74883015365971328918089051198, 4.11603475332265429479712464192, 4.80903082389288659471162524296, 5.82166186616752819680583304996, 6.64826589109098694589306355391, 7.57134226375180701430166423997, 8.006711376788452494437814339938, 8.950430765781947956289951545116

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