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Properties

Label 2-42e2-28.27-c1-0-87
Degree $2$
Conductor $1764$
Sign $0.156 + 0.987i$
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  + 1.41·2-s + 2.00·4-s − 3.37i·5-s + 2.82·8-s − 4.77i·10-s + 1.39i·13-s + 4.00·16-s − 8.15i·17-s − 6.75i·20-s − 6.41·25-s + 1.97i·26-s − 4.24·29-s + 5.65·32-s − 11.5i·34-s + 7.07·37-s + ⋯
L(s)  = 1  + 1.00·2-s + 1.00·4-s − 1.51i·5-s + 1.00·8-s − 1.51i·10-s + 0.388i·13-s + 1.00·16-s − 1.97i·17-s − 1.51i·20-s − 1.28·25-s + 0.388i·26-s − 0.787·29-s + 1.00·32-s − 1.97i·34-s + 1.16·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.156 + 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.156 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.156 + 0.987i$
Analytic conductor: \(14.0856\)
Root analytic conductor: \(3.75308\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (1567, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1764,\ (\ :1/2),\ 0.156 + 0.987i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.319611752\)
\(L(\frac12)\) \(\approx\) \(3.319611752\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 1.41T \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + 3.37iT - 5T^{2} \)
11 \( 1 - 11T^{2} \)
13 \( 1 - 1.39iT - 13T^{2} \)
17 \( 1 + 8.15iT - 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 + 4.24T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 - 7.07T + 37T^{2} \)
41 \( 1 - 6.17iT - 41T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + 4T + 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + 14.9iT - 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 - 0.579iT - 73T^{2} \)
79 \( 1 - 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + 10.9iT - 89T^{2} \)
97 \( 1 - 19.6iT - 97T^{2} \)
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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

−9.301507610632376964872642599879, −8.188470222007784164176125934729, −7.50306296606857900497622921554, −6.57851876510613705628372650611, −5.61366639427172163821020686088, −4.88312871357290656739655883051, −4.43719382837873118106220638101, −3.27540699370552373651557713586, −2.11611732093774316775310984575, −0.891598242377591644861462661617, 1.79381874791531551182182784847, 2.77277175510797967090037221933, 3.58928274412462435026470529822, 4.29822403628584107259314838131, 5.69507696326648819571209101051, 6.10321191807448411771875616533, 6.96376081244521739012639843866, 7.60130986101027903201835005902, 8.444266776689618049204060498707, 9.819309306528001724592613172803

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