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Properties

Label 2-42e2-63.59-c1-0-19
Degree $2$
Conductor $1764$
Sign $0.794 + 0.606i$
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  + (1.71 + 0.206i)3-s − 4.18·5-s + (2.91 + 0.711i)9-s − 1.42i·11-s + (0.850 + 0.491i)13-s + (−7.19 − 0.866i)15-s + (0.185 − 0.321i)17-s + (−4.30 + 2.48i)19-s − 5.75i·23-s + 12.5·25-s + (4.86 + 1.82i)27-s + (7.31 − 4.22i)29-s + (6.28 − 3.62i)31-s + (0.294 − 2.44i)33-s + (−1.73 − 3.00i)37-s + ⋯
L(s)  = 1  + (0.992 + 0.119i)3-s − 1.87·5-s + (0.971 + 0.237i)9-s − 0.429i·11-s + (0.235 + 0.136i)13-s + (−1.85 − 0.223i)15-s + (0.0449 − 0.0779i)17-s + (−0.988 + 0.570i)19-s − 1.20i·23-s + 2.50·25-s + (0.936 + 0.351i)27-s + (1.35 − 0.784i)29-s + (1.12 − 0.651i)31-s + (0.0512 − 0.425i)33-s + (−0.284 − 0.493i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.794 + 0.606i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.794 + 0.606i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.794 + 0.606i$
Analytic conductor: \(14.0856\)
Root analytic conductor: \(3.75308\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (1697, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1764,\ (\ :1/2),\ 0.794 + 0.606i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.647689601\)
\(L(\frac12)\) \(\approx\) \(1.647689601\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.71 - 0.206i)T \)
7 \( 1 \)
good5 \( 1 + 4.18T + 5T^{2} \)
11 \( 1 + 1.42iT - 11T^{2} \)
13 \( 1 + (-0.850 - 0.491i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-0.185 + 0.321i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (4.30 - 2.48i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + 5.75iT - 23T^{2} \)
29 \( 1 + (-7.31 + 4.22i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-6.28 + 3.62i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.73 + 3.00i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1.06 + 1.85i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.00 - 5.21i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4.13 + 7.16i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (4.30 + 2.48i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2.27 - 3.94i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6.50 - 3.75i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.03 - 8.71i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 10.9iT - 71T^{2} \)
73 \( 1 + (8.25 + 4.76i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.25 + 7.37i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (0.972 + 1.68i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (3.90 + 6.76i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-3.34 + 1.92i)T + (48.5 - 84.0i)T^{2} \)
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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

−8.763558899807737473711840232939, −8.446467150258650958321200319156, −7.87082353058700444268432997335, −7.07476006431389234222869630550, −6.22116384862032284824790589443, −4.56769765409827209221714649756, −4.21203492567175447773539459879, −3.34863833582913558749971172678, −2.43952757869804375458311033338, −0.67773504726193214925363301324, 1.08899242454003978017365315579, 2.66670454196793040020472467650, 3.47947607103633684848048257854, 4.23792674439153780268572252131, 4.91623347674613350939513819640, 6.61026073550280351412482024038, 7.14446803324142779729810627593, 7.979888184197476059444615663905, 8.402580394259178176919751986135, 9.098462600921621190151863269822

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