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Properties

Label 2-42e2-9.4-c1-0-20
Degree $2$
Conductor $1764$
Sign $0.313 + 0.949i$
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 + 0.124i)3-s + (−1.73 + 3.01i)5-s + (2.96 − 0.431i)9-s + (−1.25 − 2.17i)11-s + (−0.292 + 0.505i)13-s + (2.62 − 5.42i)15-s + 1.09·17-s − 5.93·19-s + (−3.19 + 5.52i)23-s + (−3.55 − 6.15i)25-s + (−5.07 + 1.11i)27-s + (0.918 + 1.59i)29-s + (−3.51 + 6.09i)31-s + (2.44 + 3.60i)33-s − 1.40·37-s + ⋯
L(s)  = 1  + (−0.997 + 0.0721i)3-s + (−0.778 + 1.34i)5-s + (0.989 − 0.143i)9-s + (−0.379 − 0.656i)11-s + (−0.0810 + 0.140i)13-s + (0.678 − 1.40i)15-s + 0.265·17-s − 1.36·19-s + (−0.665 + 1.15i)23-s + (−0.710 − 1.23i)25-s + (−0.976 + 0.214i)27-s + (0.170 + 0.295i)29-s + (−0.631 + 1.09i)31-s + (0.425 + 0.627i)33-s − 0.231·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.313 + 0.949i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.313 + 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.313 + 0.949i$
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: \((2,\ 1764,\ (\ :1/2),\ 0.313 + 0.949i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3476519064\)
\(L(\frac12)\) \(\approx\) \(0.3476519064\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.72 - 0.124i)T \)
7 \( 1 \)
good5 \( 1 + (1.73 - 3.01i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.25 + 2.17i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.292 - 0.505i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 1.09T + 17T^{2} \)
19 \( 1 + 5.93T + 19T^{2} \)
23 \( 1 + (3.19 - 5.52i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.918 - 1.59i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.51 - 6.09i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 1.40T + 37T^{2} \)
41 \( 1 + (-5.37 + 9.31i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (5.67 + 9.83i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3.76 - 6.52i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 11.6T + 53T^{2} \)
59 \( 1 + (-2.22 + 3.85i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (6.17 + 10.6i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6.33 + 10.9i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 4.93T + 71T^{2} \)
73 \( 1 - 8.71T + 73T^{2} \)
79 \( 1 + (-0.280 - 0.485i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (3.68 + 6.38i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 12.1T + 89T^{2} \)
97 \( 1 + (6.98 + 12.0i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.217237232050430273013994259126, −8.167135176280835190392007942556, −7.33740177228996824869941513501, −6.79261291213393413620606959807, −5.98722551804189655471573395658, −5.18907297628816534698443414353, −3.97686316805466555623818608643, −3.39991089046578261480340387825, −2.03741848412178840070040362449, −0.19049812709674808531589148802, 0.923916308569710567632737425637, 2.25779341363306444709129470639, 4.16673041092370958248955059272, 4.39830649077907980905875010050, 5.32698404145192458756348121595, 6.12985274750084570377972304229, 7.08650178993723346660949370236, 7.970625843578736775795026261699, 8.484846483616858177209772941595, 9.537280136241147625368058798974

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