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Properties

Label 2-42e2-63.5-c1-0-1
Degree $2$
Conductor $1764$
Sign $-0.991 + 0.129i$
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  + (−0.538 + 1.64i)3-s + (−1.21 − 2.10i)5-s + (−2.42 − 1.77i)9-s + (−2.09 − 1.21i)11-s + (4.73 + 2.73i)13-s + (4.10 − 0.866i)15-s + (1.29 + 2.23i)17-s + (0.348 + 0.201i)19-s + (−3.06 + 1.77i)23-s + (−0.440 + 0.762i)25-s + (4.21 − 3.03i)27-s + (−6.31 + 3.64i)29-s − 4.20i·31-s + (3.12 − 2.80i)33-s + (1.59 − 2.76i)37-s + ⋯
L(s)  = 1  + (−0.310 + 0.950i)3-s + (−0.542 − 0.939i)5-s + (−0.806 − 0.590i)9-s + (−0.632 − 0.365i)11-s + (1.31 + 0.758i)13-s + (1.06 − 0.223i)15-s + (0.312 + 0.542i)17-s + (0.0800 + 0.0461i)19-s + (−0.639 + 0.369i)23-s + (−0.0880 + 0.152i)25-s + (0.812 − 0.583i)27-s + (−1.17 + 0.677i)29-s − 0.754i·31-s + (0.543 − 0.487i)33-s + (0.262 − 0.454i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 + 0.129i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.991 + 0.129i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.991 + 0.129i$
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: \((2,\ 1764,\ (\ :1/2),\ -0.991 + 0.129i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1517053740\)
\(L(\frac12)\) \(\approx\) \(0.1517053740\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.538 - 1.64i)T \)
7 \( 1 \)
good5 \( 1 + (1.21 + 2.10i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (2.09 + 1.21i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-4.73 - 2.73i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1.29 - 2.23i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.348 - 0.201i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.06 - 1.77i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (6.31 - 3.64i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + 4.20iT - 31T^{2} \)
37 \( 1 + (-1.59 + 2.76i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (4.03 - 6.99i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.22 + 7.31i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 4.51T + 47T^{2} \)
53 \( 1 + (12.1 - 7.01i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + 0.155T + 59T^{2} \)
61 \( 1 - 11.8iT - 61T^{2} \)
67 \( 1 + 5.07T + 67T^{2} \)
71 \( 1 + 8.73iT - 71T^{2} \)
73 \( 1 + (7.62 - 4.40i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + 11.3T + 79T^{2} \)
83 \( 1 + (7.50 + 13.0i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (7.83 - 13.5i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (4.97 - 2.87i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.669776657272612931534784317961, −8.799958553218234310940522659596, −8.446965648776568841642739687017, −7.51721921134568018039180905488, −6.17245855651996731543593957871, −5.65479985185659401266036360813, −4.66948393773737658991528811409, −3.99704716444863618216202243607, −3.23071886908467615602256667820, −1.48552753607615977139470773773, 0.06019500833646774001427569537, 1.60587256575830190171989087926, 2.85088318971674112399917031373, 3.53561991220829477626287276086, 4.93734044079663657858710787757, 5.84562328212744836292997309381, 6.55696638618267189640088036691, 7.33751266852678382508936116365, 7.943588629740323288907660189614, 8.549412377229261934128612120190

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