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Properties

Label 2-42e2-7.2-c1-0-13
Degree $2$
Conductor $1764$
Sign $0.386 + 0.922i$
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 − 1.73i)5-s + (1 + 1.73i)11-s + 3·13-s + (−4 − 6.92i)17-s + (−0.5 + 0.866i)19-s + (4 − 6.92i)23-s + (0.500 + 0.866i)25-s − 4·29-s + (1.5 + 2.59i)31-s + (0.5 − 0.866i)37-s + 6·41-s + 11·43-s + (−3 + 5.19i)47-s + (−6 − 10.3i)53-s + 3.99·55-s + ⋯
L(s)  = 1  + (0.447 − 0.774i)5-s + (0.301 + 0.522i)11-s + 0.832·13-s + (−0.970 − 1.68i)17-s + (−0.114 + 0.198i)19-s + (0.834 − 1.44i)23-s + (0.100 + 0.173i)25-s − 0.742·29-s + (0.269 + 0.466i)31-s + (0.0821 − 0.142i)37-s + 0.937·41-s + 1.67·43-s + (−0.437 + 0.757i)47-s + (−0.824 − 1.42i)53-s + 0.539·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.386 + 0.922i$
Analytic conductor: \(14.0856\)
Root analytic conductor: \(3.75308\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (1549, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1764,\ (\ :1/2),\ 0.386 + 0.922i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.840793851\)
\(L(\frac12)\) \(\approx\) \(1.840793851\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 \)
good5 \( 1 + (-1 + 1.73i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1 - 1.73i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 3T + 13T^{2} \)
17 \( 1 + (4 + 6.92i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-4 + 6.92i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 4T + 29T^{2} \)
31 \( 1 + (-1.5 - 2.59i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.5 + 0.866i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 6T + 41T^{2} \)
43 \( 1 - 11T + 43T^{2} \)
47 \( 1 + (3 - 5.19i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (6 + 10.3i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (2 + 3.46i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3 - 5.19i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.5 + 11.2i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 10T + 71T^{2} \)
73 \( 1 + (5.5 + 9.52i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.5 + 2.59i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 2T + 83T^{2} \)
89 \( 1 + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 10T + 97T^{2} \)
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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

−9.175550666949976444772416203353, −8.598943728540689125764411132755, −7.53212569949940837434048161319, −6.73530093561808655321780487484, −5.94437176829392206325765159045, −4.88303167738140100368889216539, −4.43649725074224532532011486599, −3.08688043713126308262291844766, −1.98297348320389184406725278706, −0.74784149754909917641218120717, 1.34866671483083759308299642893, 2.49995907952739034186634807640, 3.55133581738399592637419895008, 4.30848011227964086679683880672, 5.74142686815084228301841584956, 6.13676119711765140903536923150, 6.95911482504656434009317315854, 7.87282857043256791873736466077, 8.765675855405502971298834920763, 9.340101777296431115495343362653

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