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Properties

Label 2-42e2-7.4-c3-0-14
Degree $2$
Conductor $1764$
Sign $0.947 - 0.318i$
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  + (−9.57 − 16.5i)5-s + (20.2 − 35.1i)11-s + 50.4·13-s + (−25.9 + 44.9i)17-s + (16.5 + 28.6i)19-s + (31.4 + 54.4i)23-s + (−120. + 209. i)25-s − 129.·29-s + (−121. + 209. i)31-s + (194. + 337. i)37-s + 470.·41-s − 125.·43-s + (−193. − 334. i)47-s + (−305. + 529. i)53-s − 777.·55-s + ⋯
L(s)  = 1  + (−0.855 − 1.48i)5-s + (0.556 − 0.963i)11-s + 1.07·13-s + (−0.370 + 0.641i)17-s + (0.200 + 0.346i)19-s + (0.284 + 0.493i)23-s + (−0.965 + 1.67i)25-s − 0.831·29-s + (−0.702 + 1.21i)31-s + (0.865 + 1.49i)37-s + 1.79·41-s − 0.443·43-s + (−0.599 − 1.03i)47-s + (−0.792 + 1.37i)53-s − 1.90·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.947 - 0.318i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.947 - 0.318i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.947 - 0.318i$
Analytic conductor: \(104.079\)
Root analytic conductor: \(10.2019\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1764,\ (\ :3/2),\ 0.947 - 0.318i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.483739857\)
\(L(\frac12)\) \(\approx\) \(1.483739857\)
\(L(\frac{5}{2})\) 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 \( 1 + (9.57 + 16.5i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-20.2 + 35.1i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 - 50.4T + 2.19e3T^{2} \)
17 \( 1 + (25.9 - 44.9i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-16.5 - 28.6i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-31.4 - 54.4i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + 129.T + 2.43e4T^{2} \)
31 \( 1 + (121. - 209. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-194. - 337. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 - 470.T + 6.89e4T^{2} \)
43 \( 1 + 125.T + 7.95e4T^{2} \)
47 \( 1 + (193. + 334. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (305. - 529. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-113. + 196. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (362. + 628. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (522. - 905. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 169.T + 3.57e5T^{2} \)
73 \( 1 + (-190. + 330. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-580. - 1.00e3i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 - 808.T + 5.71e5T^{2} \)
89 \( 1 + (159. + 277. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 1.13e3T + 9.12e5T^{2} \)
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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

−8.909770602364927378270403956972, −8.272989582455424032674323950498, −7.69633832351037159913306510280, −6.46031443793160947048527061404, −5.70756261181702471544744026226, −4.82533101205749247707365787929, −3.92432348123187948861290986197, −3.36056608093007440806040358563, −1.51826494139700207654903527441, −0.883982853640930134899501479969, 0.40791518830276459777036642571, 2.00032308049281848036165825004, 2.96585871650673522863396454244, 3.86004270800906453602863884828, 4.50111028689513987132087251226, 5.93504048154614034613598139814, 6.58077648138267587334328423882, 7.43056331267512512072340573163, 7.73781457399856824684967139011, 9.045372817159120731735209568557

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