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Properties

Label 2-42e2-63.41-c1-0-33
Degree $2$
Conductor $1764$
Sign $0.215 + 0.976i$
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  + (1.55 − 0.753i)3-s + (1.09 − 1.89i)5-s + (1.86 − 2.34i)9-s + (1.26 − 0.732i)11-s + (2.92 + 1.69i)13-s + (0.278 − 3.77i)15-s − 2.64·17-s − 7.94i·19-s + (3.47 + 2.00i)23-s + (0.117 + 0.203i)25-s + (1.14 − 5.06i)27-s + (−6.71 + 3.87i)29-s + (0.612 + 0.353i)31-s + (1.42 − 2.09i)33-s − 2.83·37-s + ⋯
L(s)  = 1  + (0.900 − 0.434i)3-s + (0.488 − 0.845i)5-s + (0.621 − 0.783i)9-s + (0.382 − 0.220i)11-s + (0.811 + 0.468i)13-s + (0.0720 − 0.973i)15-s − 0.640·17-s − 1.82i·19-s + (0.724 + 0.418i)23-s + (0.0234 + 0.0406i)25-s + (0.219 − 0.975i)27-s + (−1.24 + 0.719i)29-s + (0.109 + 0.0634i)31-s + (0.248 − 0.365i)33-s − 0.466·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.215 + 0.976i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.215 + 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.215 + 0.976i$
Analytic conductor: \(14.0856\)
Root analytic conductor: \(3.75308\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (293, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1764,\ (\ :1/2),\ 0.215 + 0.976i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.745264342\)
\(L(\frac12)\) \(\approx\) \(2.745264342\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.55 + 0.753i)T \)
7 \( 1 \)
good5 \( 1 + (-1.09 + 1.89i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.26 + 0.732i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.92 - 1.69i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + 2.64T + 17T^{2} \)
19 \( 1 + 7.94iT - 19T^{2} \)
23 \( 1 + (-3.47 - 2.00i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (6.71 - 3.87i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-0.612 - 0.353i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + 2.83T + 37T^{2} \)
41 \( 1 + (-3.74 + 6.48i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.27 + 2.20i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-6.27 - 10.8i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 2.79iT - 53T^{2} \)
59 \( 1 + (6.71 - 11.6i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.75 + 3.89i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.92 - 5.05i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 11.6iT - 71T^{2} \)
73 \( 1 + 4.57iT - 73T^{2} \)
79 \( 1 + (4.69 + 8.12i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1.70 - 2.95i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 9.23T + 89T^{2} \)
97 \( 1 + (-6.38 + 3.68i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.002685123675382572692503190218, −8.757038711248133901787321994691, −7.48638071015477475141502405866, −6.89904723657553553489075274231, −5.98745008050794389708459311764, −4.95678431064130645321039252777, −4.06578541418017865392097343174, −3.05172970372551678665176751986, −1.93230950273160661690461436028, −0.986391069696060367555858895770, 1.63750084683017784274879172690, 2.59556450599610140232043523554, 3.54798157397592023491458493918, 4.21942579109629317015436011410, 5.48125057807554025872933670505, 6.31335610246803650244422246190, 7.13631845649748576774146219514, 8.034071381507504468177428961475, 8.657239089662085799142572551506, 9.553669059892052297292860871269

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