Properties

Label 2-42e2-63.38-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: $\chi_{1764} (1109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1764,\ (\ :1/2),\ -0.865 + 0.501i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9781172324\)
\(L(\frac12)\) \(\approx\) \(0.9781172324\)
\(L(\frac{3}{2})\) 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

−10.02614859954483865113586720346, −8.758307800140959329013601124853, −8.334306857904077060369752073907, −7.36769584409660700372743852185, −6.75420825307106180434334723729, −5.81534388360042086893356508197, −4.44584152749110961971048610795, −3.85960812197162817749103621775, −3.06671627917794287069289321342, −2.22541130133437318826005066572, 0.33865429473411780854399324967, 1.41989455507221031046802955142, 2.60902154264884902774869888507, 4.01389168541084981343826302083, 4.41328503415412213521110946591, 5.61300707355547009152197320394, 6.52931435242141620594587723266, 7.59695627396870782478652762901, 8.015451239893741050064186452968, 8.759707737351747889685453386934

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