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Properties

Label 2-42e2-12.11-c1-0-40
Degree $2$
Conductor $1764$
Sign $0.853 + 0.521i$
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  + (−1.03 − 0.965i)2-s + (0.133 + 1.99i)4-s + 2.73i·5-s + (1.78 − 2.19i)8-s + (2.63 − 2.82i)10-s + 5.64·11-s − 1.41·13-s + (−3.96 + 0.534i)16-s − 6.19i·17-s − 4.13i·19-s + (−5.45 + 0.366i)20-s + (−5.83 − 5.45i)22-s + 5.64·23-s − 2.46·25-s + (1.46 + 1.36i)26-s + ⋯
L(s)  = 1  + (−0.730 − 0.683i)2-s + (0.0669 + 0.997i)4-s + 1.22i·5-s + (0.632 − 0.774i)8-s + (0.834 − 0.892i)10-s + 1.70·11-s − 0.392·13-s + (−0.991 + 0.133i)16-s − 1.50i·17-s − 0.947i·19-s + (−1.21 + 0.0818i)20-s + (−1.24 − 1.16i)22-s + 1.17·23-s − 0.492·25-s + (0.286 + 0.267i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.853 + 0.521i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.853 + 0.521i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.853 + 0.521i$
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: \((2,\ 1764,\ (\ :1/2),\ 0.853 + 0.521i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.292532513\)
\(L(\frac12)\) \(\approx\) \(1.292532513\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.03 + 0.965i)T \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 - 2.73iT - 5T^{2} \)
11 \( 1 - 5.64T + 11T^{2} \)
13 \( 1 + 1.41T + 13T^{2} \)
17 \( 1 + 6.19iT - 17T^{2} \)
19 \( 1 + 4.13iT - 19T^{2} \)
23 \( 1 - 5.64T + 23T^{2} \)
29 \( 1 - 0.378iT - 29T^{2} \)
31 \( 1 + 7.15iT - 31T^{2} \)
37 \( 1 + 3.46T + 37T^{2} \)
41 \( 1 + 5.26iT - 41T^{2} \)
43 \( 1 - 5.84iT - 43T^{2} \)
47 \( 1 - 2.13T + 47T^{2} \)
53 \( 1 + 0.656iT - 53T^{2} \)
59 \( 1 - 13.8T + 59T^{2} \)
61 \( 1 + 15.0T + 61T^{2} \)
67 \( 1 + 7.98iT - 67T^{2} \)
71 \( 1 - 13.9T + 71T^{2} \)
73 \( 1 - 13.0T + 73T^{2} \)
79 \( 1 - 13.8iT - 79T^{2} \)
83 \( 1 + 10.1T + 83T^{2} \)
89 \( 1 - 5.26iT - 89T^{2} \)
97 \( 1 - 9.41T + 97T^{2} \)
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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

−9.407093621369379779462739383457, −8.730049038583323389541395547436, −7.51193800375801469086996724994, −6.97427603419105827123232951336, −6.48774329476034860153253980769, −4.95169261220903042354631354674, −3.89388030153895632446224973326, −3.03661816073466834470348246859, −2.27682443552204832855126895819, −0.805491881905255210414245545849, 1.06389229011593480632931069136, 1.73707827368119040034246332322, 3.68366034886239055982321966292, 4.61196189350349385458312830306, 5.42230539401445919752675001602, 6.29505689094514045250981442218, 6.96450948451454727852621134770, 7.983253786758277592944196071244, 8.800275369951959831362223732322, 8.962312972164930986804193586848

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