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Properties

Label 2-42e2-7.4-c3-0-7
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  + (−8.32 − 14.4i)5-s + (−35.8 + 62.1i)11-s − 65.3·13-s + (45.2 − 78.3i)17-s + (−81.9 − 141. i)19-s + (39.6 + 68.6i)23-s + (−76.1 + 131. i)25-s + 43.2·29-s + (−67.8 + 117. i)31-s + (−135. − 234. i)37-s − 152.·41-s − 177.·43-s + (−22.8 − 39.5i)47-s + (−79.2 + 137. i)53-s + 1.19e3·55-s + ⋯
L(s)  = 1  + (−0.744 − 1.29i)5-s + (−0.983 + 1.70i)11-s − 1.39·13-s + (0.645 − 1.11i)17-s + (−0.989 − 1.71i)19-s + (0.359 + 0.622i)23-s + (−0.609 + 1.05i)25-s + 0.277·29-s + (−0.392 + 0.680i)31-s + (−0.601 − 1.04i)37-s − 0.579·41-s − 0.630·43-s + (−0.0708 − 0.122i)47-s + (−0.205 + 0.355i)53-s + 2.93·55-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

\(L(2)\) \(\approx\) \(0.6522006880\)
\(L(\frac12)\) \(\approx\) \(0.6522006880\)
\(L(\frac{5}{2})\) 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 \( 1 + (8.32 + 14.4i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (35.8 - 62.1i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 65.3T + 2.19e3T^{2} \)
17 \( 1 + (-45.2 + 78.3i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (81.9 + 141. i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-39.6 - 68.6i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 43.2T + 2.43e4T^{2} \)
31 \( 1 + (67.8 - 117. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (135. + 234. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 152.T + 6.89e4T^{2} \)
43 \( 1 + 177.T + 7.95e4T^{2} \)
47 \( 1 + (22.8 + 39.5i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (79.2 - 137. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (195. - 339. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (275. + 477. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (229. - 397. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 486.T + 3.57e5T^{2} \)
73 \( 1 + (287. - 497. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-334. - 578. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 76.2T + 5.71e5T^{2} \)
89 \( 1 + (-683. - 1.18e3i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 242.T + 9.12e5T^{2} \)
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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

−9.070120460341537261436500191984, −8.142824391446262611469009305015, −7.28226500625942507509372616695, −7.05712522721911299461734037314, −5.17423799210859674511449549209, −4.98760978057076154444859240757, −4.32902522818244675937542958347, −2.89104646644580571035673813878, −1.95165455319357096895911186847, −0.52230672274947979164136257489, 0.25213018043348586144227688714, 2.01182784723589419450177278172, 3.14146844966943710617663029961, 3.52889291407296043536966168517, 4.76046810695460875342373134998, 5.89003980343260765571564145771, 6.40649096676015307017207056370, 7.49295065453507026750725405003, 8.027315652851038264684884543965, 8.587892692110293566257669187512

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