Properties

Label 2-42e2-63.5-c1-0-27
Degree $2$
Conductor $1764$
Sign $-0.0820 + 0.996i$
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  + (−1.70 − 0.276i)3-s + (1.95 + 3.39i)5-s + (2.84 + 0.943i)9-s + (−3.19 − 1.84i)11-s + (−0.480 − 0.277i)13-s + (−2.41 − 6.33i)15-s + (−2.91 − 5.05i)17-s + (−4.62 − 2.66i)19-s + (−1.96 + 1.13i)23-s + (−5.16 + 8.94i)25-s + (−4.60 − 2.40i)27-s + (3.53 − 2.04i)29-s − 8.08i·31-s + (4.96 + 4.04i)33-s + (3.89 − 6.75i)37-s + ⋯
L(s)  = 1  + (−0.987 − 0.159i)3-s + (0.875 + 1.51i)5-s + (0.949 + 0.314i)9-s + (−0.964 − 0.556i)11-s + (−0.133 − 0.0769i)13-s + (−0.622 − 1.63i)15-s + (−0.707 − 1.22i)17-s + (−1.06 − 0.612i)19-s + (−0.410 + 0.237i)23-s + (−1.03 + 1.78i)25-s + (−0.886 − 0.461i)27-s + (0.656 − 0.379i)29-s − 1.45i·31-s + (0.863 + 0.703i)33-s + (0.640 − 1.11i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0820 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0820 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.0820 + 0.996i$
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: \((2,\ 1764,\ (\ :1/2),\ -0.0820 + 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6122280922\)
\(L(\frac12)\) \(\approx\) \(0.6122280922\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.70 + 0.276i)T \)
7 \( 1 \)
good5 \( 1 + (-1.95 - 3.39i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (3.19 + 1.84i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.480 + 0.277i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (2.91 + 5.05i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (4.62 + 2.66i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.96 - 1.13i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.53 + 2.04i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + 8.08iT - 31T^{2} \)
37 \( 1 + (-3.89 + 6.75i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (3.59 - 6.22i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.754 + 1.30i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 2.82T + 47T^{2} \)
53 \( 1 + (0.0415 - 0.0239i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 - 8.91T + 59T^{2} \)
61 \( 1 + 6.96iT - 61T^{2} \)
67 \( 1 - 1.17T + 67T^{2} \)
71 \( 1 - 6.71iT - 71T^{2} \)
73 \( 1 + (-3.52 + 2.03i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + 3.94T + 79T^{2} \)
83 \( 1 + (3.84 + 6.66i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-2.71 + 4.69i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (13.9 - 8.07i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.427601811171790315832540934740, −8.092971460568363935901521173589, −7.22744374033562081539765624868, −6.61646210211195465481317047337, −5.96942166128872617659933090144, −5.25857496812862306923315920719, −4.18730630258395147717739191348, −2.75338267403082263945990904419, −2.19745713125741169494829014481, −0.25952523368226119588875459943, 1.29355139693573110752912064302, 2.17918543101783057376280778154, 4.07246488941818442831289879444, 4.75336125269924910228570345035, 5.37527845751406248116821905020, 6.13099005824155333437117940700, 6.87049618329384317587543334006, 8.206191709233956866721817105984, 8.646937203563050848943430778426, 9.649501441127452333946287719560

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