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Properties

Label 2-42e2-12.11-c1-0-38
Degree $2$
Conductor $1764$
Sign $0.230 - 0.973i$
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  + (1.28 + 0.581i)2-s + (1.32 + 1.50i)4-s + (0.832 + 2.70i)8-s + 0.913·11-s + (−0.5 + 3.96i)16-s + (1.17 + 0.531i)22-s + 9.39·23-s + 5·25-s + 8.89i·29-s + (−2.95 + 4.82i)32-s − 10.5·37-s + 12i·43-s + (1.20 + 1.36i)44-s + (12.1 + 5.46i)46-s + (6.44 + 2.90i)50-s + ⋯
L(s)  = 1  + (0.911 + 0.411i)2-s + (0.661 + 0.750i)4-s + (0.294 + 0.955i)8-s + 0.275·11-s + (−0.125 + 0.992i)16-s + (0.250 + 0.113i)22-s + 1.95·23-s + 25-s + 1.65i·29-s + (−0.522 + 0.852i)32-s − 1.73·37-s + 1.82i·43-s + (0.182 + 0.206i)44-s + (1.78 + 0.806i)46-s + (0.911 + 0.411i)50-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.230 - 0.973i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.230 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.230 - 0.973i$
Analytic conductor: \(14.0856\)
Root analytic conductor: \(3.75308\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (1079, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1764,\ (\ :1/2),\ 0.230 - 0.973i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.170769980\)
\(L(\frac12)\) \(\approx\) \(3.170769980\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.28 - 0.581i)T \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 - 5T^{2} \)
11 \( 1 - 0.913T + 11T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 - 17T^{2} \)
19 \( 1 - 19T^{2} \)
23 \( 1 - 9.39T + 23T^{2} \)
29 \( 1 - 8.89iT - 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 + 10.5T + 37T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 - 12iT - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + 0.412iT - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + 61T^{2} \)
67 \( 1 + 15.8iT - 67T^{2} \)
71 \( 1 - 7.57T + 71T^{2} \)
73 \( 1 + 73T^{2} \)
79 \( 1 + 15.8iT - 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 - 89T^{2} \)
97 \( 1 + 97T^{2} \)
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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

−9.174066448912694900290115416131, −8.684153242970177146511242399687, −7.64549226793965876631078983396, −6.90426498162030719183097794959, −6.36566644885443735203957101165, −5.13545946154001093877601075863, −4.82673070729164850576513808835, −3.51956736401028740466167290689, −2.91191697271082113680917885927, −1.48428093314435314218597358849, 0.953459356125083143976812203477, 2.24579351384033417156779402665, 3.20573116126984107624550954939, 4.09029705787922518074798100278, 5.02058027669139174602392849675, 5.66450177878522552455293365162, 6.78471030594308090221585515396, 7.14016691441203699165455518370, 8.442915374994095345901444833655, 9.223594412285425047882987298543

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