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Properties

Label 2-42e2-63.16-c1-0-10
Degree $2$
Conductor $1764$
Sign $-0.678 - 0.734i$
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  + 1.73i·3-s − 2·5-s − 2.99·9-s + 4·11-s + (1.5 + 2.59i)13-s − 3.46i·15-s + (3.5 + 6.06i)17-s + (2.5 − 4.33i)19-s + 4·23-s − 25-s − 5.19i·27-s + (0.5 − 0.866i)29-s + (−1.5 + 2.59i)31-s + 6.92i·33-s + (−5.5 + 9.52i)37-s + ⋯
L(s)  = 1  + 0.999i·3-s − 0.894·5-s − 0.999·9-s + 1.20·11-s + (0.416 + 0.720i)13-s − 0.894i·15-s + (0.848 + 1.47i)17-s + (0.573 − 0.993i)19-s + 0.834·23-s − 0.200·25-s − 0.999i·27-s + (0.0928 − 0.160i)29-s + (−0.269 + 0.466i)31-s + 1.20i·33-s + (−0.904 + 1.56i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.678 - 0.734i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.678 - 0.734i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.678 - 0.734i$
Analytic conductor: \(14.0856\)
Root analytic conductor: \(3.75308\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (961, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1764,\ (\ :1/2),\ -0.678 - 0.734i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.269316971\)
\(L(\frac12)\) \(\approx\) \(1.269316971\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - 1.73iT \)
7 \( 1 \)
good5 \( 1 + 2T + 5T^{2} \)
11 \( 1 - 4T + 11T^{2} \)
13 \( 1 + (-1.5 - 2.59i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-3.5 - 6.06i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.5 + 4.33i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 4T + 23T^{2} \)
29 \( 1 + (-0.5 + 0.866i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.5 - 2.59i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (5.5 - 9.52i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (4.5 + 7.79i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.5 - 4.33i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.5 - 2.59i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.5 + 2.59i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.5 - 6.06i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.5 - 2.59i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.5 - 11.2i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 8T + 71T^{2} \)
73 \( 1 + (-3.5 - 6.06i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.5 - 7.79i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.5 + 0.866i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-7.5 + 12.9i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (8.5 - 14.7i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.521497516593288177045909215524, −8.781063623979047182552059323909, −8.309554427668049008407616523900, −7.17611409400232669824980594988, −6.39810198905636283561422763344, −5.41045646581534329094827303224, −4.43096147736352581750600604641, −3.80349427000644550610372939138, −3.12271599804072839433526380524, −1.36668733754483602015511617227, 0.53340630972311313339088385475, 1.61052072473832336317275861859, 3.13289786345080546221183174887, 3.67211396299643473085140694807, 5.04735456832686256673997989398, 5.86122912547743670037362835554, 6.78938310413208080948441769710, 7.51702111938765394695532652845, 7.967567561907726878365917710029, 8.910270908751802445870781702847

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