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Properties

Label 2-42e2-7.4-c3-0-4
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  + (4.94 + 8.57i)5-s + (−14 + 24.2i)11-s − 4.24·13-s + (−24.7 + 42.8i)17-s + (−2.82 − 4.89i)19-s + (56 + 96.9i)23-s + (13.4 − 23.3i)25-s − 154·29-s + (−16.9 + 29.3i)31-s + (10 + 17.3i)37-s + 168.·41-s − 76·43-s + (−217. − 377. i)47-s + (−266 + 460. i)53-s − 277.·55-s + ⋯
L(s)  = 1  + (0.442 + 0.766i)5-s + (−0.383 + 0.664i)11-s − 0.0905·13-s + (−0.353 + 0.611i)17-s + (−0.0341 − 0.0591i)19-s + (0.507 + 0.879i)23-s + (0.107 − 0.187i)25-s − 0.986·29-s + (−0.0983 + 0.170i)31-s + (0.0444 + 0.0769i)37-s + 0.641·41-s − 0.269·43-s + (−0.675 − 1.17i)47-s + (−0.689 + 1.19i)53-s − 0.679·55-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

\(L(2)\) \(\approx\) \(0.5403889151\)
\(L(\frac12)\) \(\approx\) \(0.5403889151\)
\(L(\frac{5}{2})\) 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 \( 1 + (-4.94 - 8.57i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (14 - 24.2i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 4.24T + 2.19e3T^{2} \)
17 \( 1 + (24.7 - 42.8i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (2.82 + 4.89i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-56 - 96.9i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + 154T + 2.43e4T^{2} \)
31 \( 1 + (16.9 - 29.3i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-10 - 17.3i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 - 168.T + 6.89e4T^{2} \)
43 \( 1 + 76T + 7.95e4T^{2} \)
47 \( 1 + (217. + 377. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (266 - 460. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (158. - 274. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (84.1 + 145. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-186 + 322. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 168T + 3.57e5T^{2} \)
73 \( 1 + (126. - 219. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-32 - 55.4i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 673.T + 5.71e5T^{2} \)
89 \( 1 + (212. + 368. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + 1.05e3T + 9.12e5T^{2} \)
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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

−9.492411124196299840479529215117, −8.621458468534215343006897584534, −7.64056544791188118377171178393, −7.02643939492257390778192696508, −6.21545553390851700507348011175, −5.40109150961620805937903122313, −4.44361686699766475993194011883, −3.39616486040457414155815898072, −2.45380084984097101435090343821, −1.55128178279533138775699662572, 0.11248875241450074052259282687, 1.17735976574454756848259589093, 2.33701948239915661536898593873, 3.34062926136797981645300060115, 4.52537459475185058496732806578, 5.21097264884838508410363675482, 5.98114603984861458464708501148, 6.88476604518415435101460314067, 7.82214649735159320536283705037, 8.590889918189456131671816310343

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