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Properties

Label 2-42e2-63.4-c1-0-31
Degree $2$
Conductor $1764$
Sign $-0.823 - 0.566i$
Analytic cond. $14.0856$
Root an. cond. $3.75308$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.5 − 0.866i)3-s − 3·5-s + (1.5 + 2.59i)9-s + 3·11-s + (−0.5 + 0.866i)13-s + (4.5 + 2.59i)15-s + (3 − 5.19i)17-s + (−2 − 3.46i)19-s − 3·23-s + 4·25-s − 5.19i·27-s + (−1.5 − 2.59i)29-s + (2.5 + 4.33i)31-s + (−4.5 − 2.59i)33-s + (−1 − 1.73i)37-s + ⋯
L(s)  = 1  + (−0.866 − 0.499i)3-s − 1.34·5-s + (0.5 + 0.866i)9-s + 0.904·11-s + (−0.138 + 0.240i)13-s + (1.16 + 0.670i)15-s + (0.727 − 1.26i)17-s + (−0.458 − 0.794i)19-s − 0.625·23-s + 0.800·25-s − 0.999i·27-s + (−0.278 − 0.482i)29-s + (0.449 + 0.777i)31-s + (−0.783 − 0.452i)33-s + (−0.164 − 0.284i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.823 - 0.566i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.823 - 0.566i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.823 - 0.566i$
Analytic conductor: \(14.0856\)
Root analytic conductor: \(3.75308\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (949, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 1764,\ (\ :1/2),\ -0.823 - 0.566i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.5 + 0.866i)T \)
7 \( 1 \)
good5 \( 1 + 3T + 5T^{2} \)
11 \( 1 - 3T + 11T^{2} \)
13 \( 1 + (0.5 - 0.866i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-3 + 5.19i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2 + 3.46i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 3T + 23T^{2} \)
29 \( 1 + (1.5 + 2.59i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2.5 - 4.33i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (1 + 1.73i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1.5 + 2.59i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.5 - 0.866i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (4.5 - 7.79i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-3 + 5.19i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.5 + 2.59i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (6.5 - 11.2i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.5 - 6.06i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 12T + 71T^{2} \)
73 \( 1 + (5 - 8.66i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (5.5 - 9.52i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (4.5 + 7.79i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-3 - 5.19i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-5.5 - 9.52i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.710206504266210235463390436096, −7.82014685789549243214842514264, −7.21044560837577064408029856653, −6.61339915518368447320487477451, −5.59320563302622552579647413090, −4.61451726972303206318621844708, −3.99649753670924468970330636455, −2.73340419430328274153309556371, −1.19679183310770055757348002303, 0, 1.47150915248539676189471536097, 3.45356563246760492536841584847, 3.93881560402874991141100323373, 4.68518777546796800208285697480, 5.85029034320608533759182380018, 6.40918126890972494567761537091, 7.46808525755896647109077909280, 8.117051941293410790627888932049, 8.959763530829391837183297999517

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