Properties

Label 2-42e2-12.11-c1-0-21
Degree $2$
Conductor $1764$
Sign $0.577 - 0.816i$
Analytic cond. $14.0856$
Root an. cond. $3.75308$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.41i·2-s − 2.00·4-s + 4.24i·5-s + 2.82i·8-s + 6·10-s + 6·13-s + 4.00·16-s − 4.24i·17-s − 8.48i·20-s − 12.9·25-s − 8.48i·26-s + 9.89i·29-s − 5.65i·32-s − 6·34-s − 2·37-s + ⋯
L(s)  = 1  − 0.999i·2-s − 1.00·4-s + 1.89i·5-s + 1.00i·8-s + 1.89·10-s + 1.66·13-s + 1.00·16-s − 1.02i·17-s − 1.89i·20-s − 2.59·25-s − 1.66i·26-s + 1.83i·29-s − 1.00i·32-s − 1.02·34-s − 0.328·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.577 - 0.816i$
Analytic conductor: \(14.0856\)
Root analytic conductor: \(3.75308\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (1079, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1764,\ (\ :1/2),\ 0.577 - 0.816i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.357377706\)
\(L(\frac12)\) \(\approx\) \(1.357377706\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 1.41iT \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 - 4.24iT - 5T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 - 6T + 13T^{2} \)
17 \( 1 + 4.24iT - 17T^{2} \)
19 \( 1 - 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 - 9.89iT - 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 + 2T + 37T^{2} \)
41 \( 1 - 12.7iT - 41T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 7.07iT - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 12T + 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 6T + 73T^{2} \)
79 \( 1 - 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + 4.24iT - 89T^{2} \)
97 \( 1 + 18T + 97T^{2} \)
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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

−9.652173553832694733360080910155, −8.801394743501445615550369381552, −7.916842880453020762746270948022, −6.98576482890637229015149049786, −6.26884477414532737926062521435, −5.31136998390671865165742300088, −4.03822964536088680837124253099, −3.24958931687781157788739063697, −2.71113413995048840570646202076, −1.40433213360464605806209832799, 0.56543345258875710026540621223, 1.64625240467082765857055783044, 3.89160305151847577774218133684, 4.16319772384089492977424346533, 5.37875314578908481358825924015, 5.77045337030886139303586056636, 6.66046356916649474593699749730, 7.937538777983804333700619065598, 8.358622308332705733988312252279, 8.873119540056533245525347234187

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