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Properties

Label 2-42e2-7.4-c3-0-39
Degree $2$
Conductor $1764$
Sign $0.605 + 0.795i$
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 + 5.19i)5-s + (18 − 31.1i)11-s + 62·13-s + (57 − 98.7i)17-s + (38 + 65.8i)19-s + (−12 − 20.7i)23-s + (44.5 − 77.0i)25-s − 54·29-s + (56 − 96.9i)31-s + (89 + 154. i)37-s − 378·41-s − 172·43-s + (−96 − 166. i)47-s + (−201 + 348. i)53-s + 216·55-s + ⋯
L(s)  = 1  + (0.268 + 0.464i)5-s + (0.493 − 0.854i)11-s + 1.32·13-s + (0.813 − 1.40i)17-s + (0.458 + 0.794i)19-s + (−0.108 − 0.188i)23-s + (0.355 − 0.616i)25-s − 0.345·29-s + (0.324 − 0.561i)31-s + (0.395 + 0.684i)37-s − 1.43·41-s − 0.609·43-s + (−0.297 − 0.516i)47-s + (−0.520 + 0.902i)53-s + 0.529·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.605 + 0.795i$
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: \((2,\ 1764,\ (\ :3/2),\ 0.605 + 0.795i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.537984225\)
\(L(\frac12)\) \(\approx\) \(2.537984225\)
\(L(\frac{5}{2})\) 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 \( 1 + (-3 - 5.19i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-18 + 31.1i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 - 62T + 2.19e3T^{2} \)
17 \( 1 + (-57 + 98.7i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-38 - 65.8i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (12 + 20.7i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + 54T + 2.43e4T^{2} \)
31 \( 1 + (-56 + 96.9i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-89 - 154. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 378T + 6.89e4T^{2} \)
43 \( 1 + 172T + 7.95e4T^{2} \)
47 \( 1 + (96 + 166. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (201 - 348. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-198 + 342. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (127 + 219. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-506 + 876. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 840T + 3.57e5T^{2} \)
73 \( 1 + (445 - 770. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (40 + 69.2i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 - 108T + 5.71e5T^{2} \)
89 \( 1 + (819 + 1.41e3i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 1.01e3T + 9.12e5T^{2} \)
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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

−8.721452343943278549383145304237, −8.152136421632625136733224169077, −7.17816029611407992122616799436, −6.30815119834359154151319725987, −5.78773001286347016533481159173, −4.74165644082363399631367881562, −3.53044786399619899466539462131, −3.02855194034514112733958211792, −1.61695183772888529540617440184, −0.59878550989175080128156856499, 1.12547451513669133892225638442, 1.75236900299777579815509179670, 3.24262779659524419082455593518, 4.02088769281063797096125660513, 5.01606429151151892555810103087, 5.82410703443406472023309587185, 6.61889124880372168107656230822, 7.47029990351427067753005347970, 8.441200011104557433124453276930, 8.949089067676813534953711212213

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