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Properties

Label 2-42e2-7.2-c3-0-34
Degree $2$
Conductor $1764$
Sign $0.701 + 0.712i$
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 − 15.5i)5-s + (18 + 31.1i)11-s + 10·13-s + (−9 − 15.5i)17-s + (−50 + 86.6i)19-s + (36 − 62.3i)23-s + (−99.5 − 172. i)25-s + 234·29-s + (−8 − 13.8i)31-s + (113 − 195. i)37-s + 90·41-s + 452·43-s + (−216 + 374. i)47-s + (207 + 358. i)53-s + 648·55-s + ⋯
L(s)  = 1  + (0.804 − 1.39i)5-s + (0.493 + 0.854i)11-s + 0.213·13-s + (−0.128 − 0.222i)17-s + (−0.603 + 1.04i)19-s + (0.326 − 0.565i)23-s + (−0.796 − 1.37i)25-s + 1.49·29-s + (−0.0463 − 0.0802i)31-s + (0.502 − 0.869i)37-s + 0.342·41-s + 1.60·43-s + (−0.670 + 1.16i)47-s + (0.536 + 0.929i)53-s + 1.58·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.701 + 0.712i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.701 + 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.701 + 0.712i$
Analytic conductor: \(104.079\)
Root analytic conductor: \(10.2019\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (1549, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1764,\ (\ :3/2),\ 0.701 + 0.712i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.825481394\)
\(L(\frac12)\) \(\approx\) \(2.825481394\)
\(L(\frac{5}{2})\) 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 \( 1 + (-9 + 15.5i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (-18 - 31.1i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 - 10T + 2.19e3T^{2} \)
17 \( 1 + (9 + 15.5i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (50 - 86.6i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-36 + 62.3i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 - 234T + 2.43e4T^{2} \)
31 \( 1 + (8 + 13.8i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-113 + 195. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 - 90T + 6.89e4T^{2} \)
43 \( 1 - 452T + 7.95e4T^{2} \)
47 \( 1 + (216 - 374. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-207 - 358. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (-342 - 592. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-211 + 365. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (166 + 287. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 360T + 3.57e5T^{2} \)
73 \( 1 + (-13 - 22.5i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (256 - 443. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + 1.18e3T + 5.71e5T^{2} \)
89 \( 1 + (-315 + 545. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
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

−8.872689747802781363062763969872, −8.271076886378571913549930628921, −7.27405864455904072931175606736, −6.24187973631704266964349974091, −5.65188642908937928954359213343, −4.59248429739517556872411255422, −4.19164748514977124139850659722, −2.58795128738226286116707429988, −1.61247760154480089869499718296, −0.77059212253266433876616503756, 0.887450793374668581187595244892, 2.23071792738050317490231656841, 2.96128551253210483150595753623, 3.84795732219571888673878988638, 5.06782069272426223442837093718, 6.09921993341445797875673522651, 6.53918186103063353432003556766, 7.22266635615765697553398573165, 8.355825540664815877638017455014, 9.037960831042896969975298680303

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