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Properties

Label 2-42e2-9.4-c1-0-17
Degree $2$
Conductor $1764$
Sign $-0.0384 - 0.999i$
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  + (1.72 + 0.117i)3-s + (−1.95 + 3.37i)5-s + (2.97 + 0.406i)9-s + (2.13 + 3.69i)11-s + (2.71 − 4.69i)13-s + (−3.76 + 5.60i)15-s + 4.98·17-s − 0.444·19-s + (−4.10 + 7.11i)23-s + (−5.10 − 8.84i)25-s + (5.08 + 1.05i)27-s + (1.33 + 2.31i)29-s + (−0.425 + 0.737i)31-s + (3.25 + 6.63i)33-s − 9.80·37-s + ⋯
L(s)  = 1  + (0.997 + 0.0679i)3-s + (−0.872 + 1.51i)5-s + (0.990 + 0.135i)9-s + (0.643 + 1.11i)11-s + (0.752 − 1.30i)13-s + (−0.972 + 1.44i)15-s + 1.20·17-s − 0.101·19-s + (−0.856 + 1.48i)23-s + (−1.02 − 1.76i)25-s + (0.979 + 0.202i)27-s + (0.248 + 0.430i)29-s + (−0.0765 + 0.132i)31-s + (0.566 + 1.15i)33-s − 1.61·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0384 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0384 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.0384 - 0.999i$
Analytic conductor: \(14.0856\)
Root analytic conductor: \(3.75308\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (589, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1764,\ (\ :1/2),\ -0.0384 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.240789369\)
\(L(\frac12)\) \(\approx\) \(2.240789369\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.72 - 0.117i)T \)
7 \( 1 \)
good5 \( 1 + (1.95 - 3.37i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-2.13 - 3.69i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.71 + 4.69i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 4.98T + 17T^{2} \)
19 \( 1 + 0.444T + 19T^{2} \)
23 \( 1 + (4.10 - 7.11i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.33 - 2.31i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.425 - 0.737i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 9.80T + 37T^{2} \)
41 \( 1 + (-2.69 + 4.66i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.31 - 7.46i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (1.74 + 3.02i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 5.67T + 53T^{2} \)
59 \( 1 + (-0.947 + 1.64i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.62 - 9.74i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.29 - 2.24i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 0.141T + 71T^{2} \)
73 \( 1 + 10.3T + 73T^{2} \)
79 \( 1 + (-0.204 - 0.354i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2.08 - 3.60i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 2.41T + 89T^{2} \)
97 \( 1 + (-3.80 - 6.58i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.639321480807522555052364367496, −8.550715081606594038048010045684, −7.70190964073642747635374372044, −7.42364055208966536204061839148, −6.59058325678897149153521624154, −5.48727816736716107992933881267, −4.07811590815845993854533874007, −3.50523667325632084415740773053, −2.87892458708111954180762380435, −1.57427816781091623897363848400, 0.805366651411057726545397948764, 1.79190034671896737678806748148, 3.37543357172607721950996167581, 4.01727373821955446283900933802, 4.65950621900562243690417841505, 5.88814268739195771693314602930, 6.81321233104397493104537101372, 7.922356512050555156653088702893, 8.384602078225340348759213915027, 8.902702152462576387058066320612

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