Properties

Label 2-42e2-21.5-c3-0-14
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  + (4.36 − 7.56i)5-s + (7.60 − 4.39i)11-s + 11.8i·13-s + (−22.2 − 38.6i)17-s + (−10.0 − 5.82i)19-s + (123. + 71.3i)23-s + (24.3 + 42.1i)25-s + 234. i·29-s + (−252. + 145. i)31-s + (44.4 − 76.9i)37-s − 145.·41-s + 144.·43-s + (120. − 208. i)47-s + (263. − 152. i)53-s − 76.7i·55-s + ⋯
L(s)  = 1  + (0.390 − 0.676i)5-s + (0.208 − 0.120i)11-s + 0.252i·13-s + (−0.318 − 0.550i)17-s + (−0.121 − 0.0703i)19-s + (1.11 + 0.646i)23-s + (0.194 + 0.337i)25-s + 1.49i·29-s + (−1.46 + 0.845i)31-s + (0.197 − 0.342i)37-s − 0.555·41-s + 0.512·43-s + (0.372 − 0.646i)47-s + (0.683 − 0.394i)53-s − 0.188i·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: $\chi_{1764} (1097, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1764,\ (\ :3/2),\ 0.879 - 0.475i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.143791609\)
\(L(\frac12)\) \(\approx\) \(2.143791609\)
\(L(\frac{5}{2})\) 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 \( 1 + (-4.36 + 7.56i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (-7.60 + 4.39i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 - 11.8iT - 2.19e3T^{2} \)
17 \( 1 + (22.2 + 38.6i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (10.0 + 5.82i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-123. - 71.3i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 234. iT - 2.43e4T^{2} \)
31 \( 1 + (252. - 145. i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-44.4 + 76.9i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + 145.T + 6.89e4T^{2} \)
43 \( 1 - 144.T + 7.95e4T^{2} \)
47 \( 1 + (-120. + 208. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-263. + 152. i)T + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (3.54 + 6.13i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-149. - 86.4i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-243. - 421. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 653. iT - 3.57e5T^{2} \)
73 \( 1 + (-99.0 + 57.1i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (147. - 255. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 - 877.T + 5.71e5T^{2} \)
89 \( 1 + (710. - 1.23e3i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + 738. iT - 9.12e5T^{2} \)
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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

−9.059139883696867233337536548794, −8.454695325533920575792660670617, −7.19268282918731129543572587429, −6.83379644521972623826272396090, −5.42847702879305349428366057634, −5.19514755016427721622868450674, −4.01113743551705402978379722974, −3.04985376005783781024106875132, −1.81407780342717247008407190750, −0.909371453132971143041610003519, 0.54201512326477246297748584260, 1.94809092969427785084304964161, 2.76823642912253707922444395478, 3.82850513527087689115015641045, 4.72875021762971372051075784824, 5.83629408218099653283644851074, 6.42051759610777662986250077451, 7.24113069794391429106322873773, 8.050856474725547189179735768926, 8.949986418311564974955841596278

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