Properties

Label 2-42e2-7.4-c1-0-2
Degree $2$
Conductor $1764$
Sign $-0.900 - 0.435i$
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  + (1.87 + 3.24i)5-s + (−2.64 + 4.58i)11-s − 4.24·13-s + (1.87 − 3.24i)17-s + (−1.41 − 2.44i)19-s + (2.64 + 4.58i)23-s + (−4.5 + 7.79i)25-s − 5.29·29-s + (−4.24 + 7.34i)31-s + (−2 − 3.46i)37-s − 3.74·41-s + 8·43-s + (−3.74 − 6.48i)47-s + (5.29 − 9.16i)53-s − 19.7·55-s + ⋯
L(s)  = 1  + (0.836 + 1.44i)5-s + (−0.797 + 1.38i)11-s − 1.17·13-s + (0.453 − 0.785i)17-s + (−0.324 − 0.561i)19-s + (0.551 + 0.955i)23-s + (−0.900 + 1.55i)25-s − 0.982·29-s + (−0.762 + 1.31i)31-s + (−0.328 − 0.569i)37-s − 0.584·41-s + 1.21·43-s + (−0.545 − 0.945i)47-s + (0.726 − 1.25i)53-s − 2.66·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.900 - 0.435i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.900 - 0.435i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.900 - 0.435i$
Analytic conductor: \(14.0856\)
Root analytic conductor: \(3.75308\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1764,\ (\ :1/2),\ -0.900 - 0.435i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.163504296\)
\(L(\frac12)\) \(\approx\) \(1.163504296\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 \)
good5 \( 1 + (-1.87 - 3.24i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (2.64 - 4.58i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 4.24T + 13T^{2} \)
17 \( 1 + (-1.87 + 3.24i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.41 + 2.44i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.64 - 4.58i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 5.29T + 29T^{2} \)
31 \( 1 + (4.24 - 7.34i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (2 + 3.46i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 3.74T + 41T^{2} \)
43 \( 1 - 8T + 43T^{2} \)
47 \( 1 + (3.74 + 6.48i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-5.29 + 9.16i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.74 - 6.48i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.94 + 8.57i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6 - 10.3i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 15.8T + 71T^{2} \)
73 \( 1 + (0.707 - 1.22i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2 - 3.46i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 14.9T + 83T^{2} \)
89 \( 1 + (1.87 + 3.24i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 9.89T + 97T^{2} \)
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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

−9.682513664775424893118931261714, −9.202985630599059943699074851377, −7.67664629832199405376633726558, −7.17081295057720642808917320311, −6.75254827760707890800869045270, −5.41258519838945539510112503802, −5.03096923672120914097524601941, −3.55200074557467060016562719788, −2.58047197579057400473404190807, −1.98572222831471222261527423522, 0.39785779268200642485907593191, 1.68888675424938069892508832354, 2.76860850233280735045753507532, 4.07032832810425596053669280664, 5.02895046632153612596160483345, 5.64361084111887226120894121544, 6.24621944347607045113636959032, 7.67661542037800690703590471524, 8.202930422434341492777751498736, 9.029981955011173011136054600616

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