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Properties

Label 2-42e2-63.5-c1-0-11
Degree $2$
Conductor $1764$
Sign $0.886 - 0.462i$
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  + (−1.06 + 1.36i)3-s + (−0.349 − 0.605i)5-s + (−0.721 − 2.91i)9-s + (0.229 + 0.132i)11-s + (−1.13 − 0.657i)13-s + (1.19 + 0.169i)15-s + (1.86 + 3.22i)17-s + (0.382 + 0.220i)19-s + (−4.29 + 2.48i)23-s + (2.25 − 3.90i)25-s + (4.74 + 2.12i)27-s + (−0.273 + 0.157i)29-s − 5.60i·31-s + (−0.426 + 0.171i)33-s + (−0.351 + 0.608i)37-s + ⋯
L(s)  = 1  + (−0.616 + 0.787i)3-s + (−0.156 − 0.270i)5-s + (−0.240 − 0.970i)9-s + (0.0692 + 0.0399i)11-s + (−0.315 − 0.182i)13-s + (0.309 + 0.0437i)15-s + (0.452 + 0.783i)17-s + (0.0877 + 0.0506i)19-s + (−0.896 + 0.517i)23-s + (0.451 − 0.781i)25-s + (0.912 + 0.408i)27-s + (−0.0507 + 0.0292i)29-s − 1.00i·31-s + (−0.0741 + 0.0298i)33-s + (−0.0577 + 0.0999i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.886 - 0.462i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.886 - 0.462i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.886 - 0.462i$
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: \((2,\ 1764,\ (\ :1/2),\ 0.886 - 0.462i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.221071227\)
\(L(\frac12)\) \(\approx\) \(1.221071227\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.06 - 1.36i)T \)
7 \( 1 \)
good5 \( 1 + (0.349 + 0.605i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.229 - 0.132i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.13 + 0.657i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1.86 - 3.22i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.382 - 0.220i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (4.29 - 2.48i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.273 - 0.157i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + 5.60iT - 31T^{2} \)
37 \( 1 + (0.351 - 0.608i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-5.39 + 9.34i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.73 - 6.46i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 7.00T + 47T^{2} \)
53 \( 1 + (-8.51 + 4.91i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 - 13.4T + 59T^{2} \)
61 \( 1 - 5.65iT - 61T^{2} \)
67 \( 1 + 5.94T + 67T^{2} \)
71 \( 1 - 13.4iT - 71T^{2} \)
73 \( 1 + (-6.66 + 3.84i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 - 1.39T + 79T^{2} \)
83 \( 1 + (-3.72 - 6.45i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (5.59 - 9.68i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-9.18 + 5.30i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.497197495438710010650365276853, −8.655917249081487395873628762598, −7.86730008845605076118859323051, −6.88927906218237472635858698523, −5.87469775297858158330666892490, −5.41696543137663865307106217714, −4.25781286346536516534327027338, −3.79831118418763754728195395259, −2.43058921368548109140922627217, −0.78227826516597763688419348761, 0.77974997205874238154881648764, 2.09933947743555807307715312117, 3.11265401102460310356583558868, 4.40035734869422157261677206278, 5.30827590087902491369571606410, 6.05287843913856321241231978838, 6.99773234521987824029692641115, 7.41965055533702294854602617624, 8.310155657990479607455503402387, 9.194520731742051067264317875747

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