Properties

Label 2-42e2-12.11-c1-0-25
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 − 1.41i·5-s + 2.82i·8-s − 2.00·10-s + 4·13-s + 4.00·16-s + 7.07i·17-s + 2.82i·20-s + 2.99·25-s − 5.65i·26-s + 9.89i·29-s − 5.65i·32-s + 10.0·34-s + 2·37-s + ⋯
L(s)  = 1  − 0.999i·2-s − 1.00·4-s − 0.632i·5-s + 1.00i·8-s − 0.632·10-s + 1.10·13-s + 1.00·16-s + 1.71i·17-s + 0.632i·20-s + 0.599·25-s − 1.10i·26-s + 1.83i·29-s − 1.00i·32-s + 1.71·34-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

\(L(1)\) \(\approx\) \(1.561206045\)
\(L(\frac12)\) \(\approx\) \(1.561206045\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.41iT - 5T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 - 4T + 13T^{2} \)
17 \( 1 - 7.07iT - 17T^{2} \)
19 \( 1 - 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 - 9.89iT - 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 - 2T + 37T^{2} \)
41 \( 1 + 1.41iT - 41T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + 7.07iT - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 10T + 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 16T + 73T^{2} \)
79 \( 1 - 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + 18.3iT - 89T^{2} \)
97 \( 1 + 8T + 97T^{2} \)
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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

−9.083559597761376916998406488775, −8.604615948543984806954555397492, −8.031822174114039517254103200728, −6.69117887297242671851382966503, −5.72250297994792867685532902065, −4.93933073014641451114900035031, −3.96120476397456095112552159462, −3.31426668407820043049485753548, −1.90683177118357553204744260386, −1.03503149258991882045679211844, 0.78090895439012213134471147038, 2.63994859600334458191895790052, 3.67887499882034198321008981467, 4.60221345991105671705217530867, 5.53493084200743304160227091114, 6.34394468186259511321388436319, 6.96859277718540326727871250078, 7.75092787742152422470158020981, 8.477249383934390045871409295796, 9.353836263512789625928165935555

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