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Properties

Label 2-42e2-28.11-c0-0-3
Degree $2$
Conductor $1764$
Sign $0.947 - 0.318i$
Analytic cond. $0.880350$
Root an. cond. $0.938270$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 + 0.866i)2-s + (−0.499 + 0.866i)4-s + (−0.707 − 1.22i)5-s − 0.999·8-s + (0.707 − 1.22i)10-s + 1.41·13-s + (−0.5 − 0.866i)16-s + (0.707 − 1.22i)17-s + 1.41·20-s + (−0.499 + 0.866i)25-s + (0.707 + 1.22i)26-s + 2·29-s + (0.499 − 0.866i)32-s + 1.41·34-s + (0.707 + 1.22i)40-s − 1.41·41-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)2-s + (−0.499 + 0.866i)4-s + (−0.707 − 1.22i)5-s − 0.999·8-s + (0.707 − 1.22i)10-s + 1.41·13-s + (−0.5 − 0.866i)16-s + (0.707 − 1.22i)17-s + 1.41·20-s + (−0.499 + 0.866i)25-s + (0.707 + 1.22i)26-s + 2·29-s + (0.499 − 0.866i)32-s + 1.41·34-s + (0.707 + 1.22i)40-s − 1.41·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.947 - 0.318i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.947 - 0.318i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.947 - 0.318i$
Analytic conductor: \(0.880350\)
Root analytic conductor: \(0.938270\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (1243, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1764,\ (\ :0),\ 0.947 - 0.318i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.261647525\)
\(L(\frac12)\) \(\approx\) \(1.261647525\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 - 0.866i)T \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (0.707 + 1.22i)T + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
13 \( 1 - 1.41T + T^{2} \)
17 \( 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.5 - 0.866i)T^{2} \)
29 \( 1 - 2T + T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 + (-0.5 + 0.866i)T^{2} \)
41 \( 1 + 1.41T + T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 + (-0.5 - 0.866i)T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (-0.707 - 1.22i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (0.707 + 1.22i)T + (-0.5 + 0.866i)T^{2} \)
97 \( 1 + 1.41T + T^{2} \)
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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

−9.136190418357849106709943878777, −8.495819016311463848801226691263, −8.092738051793060259193921884956, −7.14269190828357893965208786042, −6.30454899389750259161759559018, −5.34317484567829716724825689419, −4.73360561118579464631616814025, −3.91585588955959366340224619574, −3.01514990351060387202963401053, −0.982969382313750927505557926353, 1.38668787631605619845323702856, 2.75577738502592081834950365040, 3.53483273969920960850348035043, 4.07240218186495544181797942461, 5.30536469062082443723078861596, 6.31111624950802274783630540556, 6.75465938678443711716837449106, 8.120037079959897486396805886609, 8.546382262865481953289779136914, 9.815352260149543897585006159887

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