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Properties

Label 2-42e2-63.5-c1-0-7
Degree $2$
Conductor $1764$
Sign $-0.865 - 0.501i$
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  + (−1.03 + 1.38i)3-s + (2.09 + 3.62i)5-s + (−0.841 − 2.87i)9-s + (1.23 + 0.711i)11-s + (0.850 + 0.491i)13-s + (−7.19 − 0.866i)15-s + (0.185 + 0.321i)17-s + (4.30 + 2.48i)19-s + (−4.98 + 2.87i)23-s + (−6.26 + 10.8i)25-s + (4.86 + 1.82i)27-s + (7.31 − 4.22i)29-s + 7.25i·31-s + (−2.26 + 0.968i)33-s + (−1.73 + 3.00i)37-s + ⋯
L(s)  = 1  + (−0.599 + 0.800i)3-s + (0.936 + 1.62i)5-s + (−0.280 − 0.959i)9-s + (0.371 + 0.214i)11-s + (0.235 + 0.136i)13-s + (−1.85 − 0.223i)15-s + (0.0449 + 0.0779i)17-s + (0.988 + 0.570i)19-s + (−1.04 + 0.600i)23-s + (−1.25 + 2.17i)25-s + (0.936 + 0.351i)27-s + (1.35 − 0.784i)29-s + 1.30i·31-s + (−0.394 + 0.168i)33-s + (−0.284 + 0.493i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.865 - 0.501i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.865 - 0.501i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.865 - 0.501i$
Analytic conductor: \(14.0856\)
Root analytic conductor: \(3.75308\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (509, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1764,\ (\ :1/2),\ -0.865 - 0.501i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.559334072\)
\(L(\frac12)\) \(\approx\) \(1.559334072\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.03 - 1.38i)T \)
7 \( 1 \)
good5 \( 1 + (-2.09 - 3.62i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.23 - 0.711i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.850 - 0.491i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-0.185 - 0.321i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-4.30 - 2.48i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (4.98 - 2.87i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-7.31 + 4.22i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 7.25iT - 31T^{2} \)
37 \( 1 + (1.73 - 3.00i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-1.06 + 1.85i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.00 - 5.21i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 8.27T + 47T^{2} \)
53 \( 1 + (-4.30 + 2.48i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + 4.55T + 59T^{2} \)
61 \( 1 + 7.51iT - 61T^{2} \)
67 \( 1 + 10.0T + 67T^{2} \)
71 \( 1 + 10.9iT - 71T^{2} \)
73 \( 1 + (-8.25 + 4.76i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + 8.51T + 79T^{2} \)
83 \( 1 + (0.972 + 1.68i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (3.90 - 6.76i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-3.34 + 1.92i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.899376872875043697342492942071, −9.205543333171792696667518604250, −7.985229061825802521061805025435, −6.95938817250930596367496933790, −6.30136519234505649677113887871, −5.79951498345039345631874080407, −4.77848218388479372066113373416, −3.60082323753524060865860496036, −2.98212752046885418324499931398, −1.63410573822088408946638619649, 0.66541357790720846029741973637, 1.48290345381601732344853640659, 2.56196874674674473713318924362, 4.25562738277625370599972559002, 5.06966810895571163797027260322, 5.74950665314501910855439296850, 6.34033707401947072456475934693, 7.38721821406884882416795527687, 8.324603117210628370475228624588, 8.828162944826073610405414205114

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