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Properties

Label 2-42e2-63.20-c1-0-14
Degree $2$
Conductor $1764$
Sign $0.966 - 0.257i$
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  + (0.240 − 1.71i)3-s + (1.48 + 2.57i)5-s + (−2.88 − 0.825i)9-s + (4.09 + 2.36i)11-s + (3.54 − 2.04i)13-s + (4.76 − 1.92i)15-s − 1.67·17-s + 4.91i·19-s + (−4.25 + 2.45i)23-s + (−1.91 + 3.30i)25-s + (−2.11 + 4.74i)27-s + (0.238 + 0.137i)29-s + (1.38 − 0.801i)31-s + (5.04 − 6.45i)33-s + 3.39·37-s + ⋯
L(s)  = 1  + (0.138 − 0.990i)3-s + (0.664 + 1.15i)5-s + (−0.961 − 0.275i)9-s + (1.23 + 0.712i)11-s + (0.981 − 0.566i)13-s + (1.23 − 0.497i)15-s − 0.405·17-s + 1.12i·19-s + (−0.886 + 0.511i)23-s + (−0.382 + 0.661i)25-s + (−0.406 + 0.913i)27-s + (0.0442 + 0.0255i)29-s + (0.249 − 0.143i)31-s + (0.877 − 1.12i)33-s + 0.557·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.966 - 0.257i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.966 - 0.257i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.966 - 0.257i$
Analytic conductor: \(14.0856\)
Root analytic conductor: \(3.75308\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (1469, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1764,\ (\ :1/2),\ 0.966 - 0.257i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.149243141\)
\(L(\frac12)\) \(\approx\) \(2.149243141\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.240 + 1.71i)T \)
7 \( 1 \)
good5 \( 1 + (-1.48 - 2.57i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-4.09 - 2.36i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-3.54 + 2.04i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + 1.67T + 17T^{2} \)
19 \( 1 - 4.91iT - 19T^{2} \)
23 \( 1 + (4.25 - 2.45i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.238 - 0.137i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.38 + 0.801i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 3.39T + 37T^{2} \)
41 \( 1 + (-3.55 - 6.15i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-5.22 + 9.05i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (5.49 - 9.52i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 0.816iT - 53T^{2} \)
59 \( 1 + (-1.37 - 2.38i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6.23 - 3.60i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.80 + 10.0i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 10.4iT - 71T^{2} \)
73 \( 1 - 15.7iT - 73T^{2} \)
79 \( 1 + (-6.15 + 10.6i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-4.03 + 6.99i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 9.21T + 89T^{2} \)
97 \( 1 + (7.00 + 4.04i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.369133717291594664195928042950, −8.415304264989967632449968579220, −7.63983338393920489204726658921, −6.86307028280328744585584273977, −6.16539594909482575966781698001, −5.80823639715211432140462102878, −4.13008377863233073518065922579, −3.22413353982272147370100072583, −2.18983849762186635463974893560, −1.32989334774304490716643011722, 0.889098812284665070371060728257, 2.20870753565220419705889010343, 3.58534832688984300038242880522, 4.31327235963794295675475435339, 5.04341835045637725248711697949, 6.01779941415891322276067102073, 6.53306779154464996961205903676, 8.078132920624342065840386359835, 8.934289531646298735797739722219, 9.016164567127436056223344267670

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