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Properties

Label 2-42e2-28.27-c1-0-53
Degree $2$
Conductor $1764$
Sign $0.997 - 0.0650i$
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  + (−0.250 + 1.39i)2-s + (−1.87 − 0.697i)4-s − 3.39i·5-s + (1.44 − 2.43i)8-s + (4.72 + 0.850i)10-s + 1.94i·11-s + 5.34i·13-s + (3.02 + 2.61i)16-s + 1.74i·17-s + 6.27·19-s + (−2.36 + 6.35i)20-s + (−2.70 − 0.486i)22-s − 8.39i·23-s − 6.50·25-s + (−7.43 − 1.33i)26-s + ⋯
L(s)  = 1  + (−0.177 + 0.984i)2-s + (−0.937 − 0.348i)4-s − 1.51i·5-s + (0.509 − 0.860i)8-s + (1.49 + 0.268i)10-s + 0.585i·11-s + 1.48i·13-s + (0.756 + 0.653i)16-s + 0.423i·17-s + 1.43·19-s + (−0.529 + 1.42i)20-s + (−0.576 − 0.103i)22-s − 1.75i·23-s − 1.30·25-s + (−1.45 − 0.262i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0650i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 - 0.0650i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.997 - 0.0650i$
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: \((2,\ 1764,\ (\ :1/2),\ 0.997 - 0.0650i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.373639311\)
\(L(\frac12)\) \(\approx\) \(1.373639311\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.250 - 1.39i)T \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + 3.39iT - 5T^{2} \)
11 \( 1 - 1.94iT - 11T^{2} \)
13 \( 1 - 5.34iT - 13T^{2} \)
17 \( 1 - 1.74iT - 17T^{2} \)
19 \( 1 - 6.27T + 19T^{2} \)
23 \( 1 + 8.39iT - 23T^{2} \)
29 \( 1 + 4.18T + 29T^{2} \)
31 \( 1 - 4.59T + 31T^{2} \)
37 \( 1 + 1.78T + 37T^{2} \)
41 \( 1 + 1.32iT - 41T^{2} \)
43 \( 1 + 3.27iT - 43T^{2} \)
47 \( 1 - 8.22T + 47T^{2} \)
53 \( 1 - 11.8T + 53T^{2} \)
59 \( 1 - 0.507T + 59T^{2} \)
61 \( 1 + 7.63iT - 61T^{2} \)
67 \( 1 + 16.1iT - 67T^{2} \)
71 \( 1 - 1.11iT - 71T^{2} \)
73 \( 1 - 3.06iT - 73T^{2} \)
79 \( 1 + 6.77iT - 79T^{2} \)
83 \( 1 - 5.39T + 83T^{2} \)
89 \( 1 + 10.1iT - 89T^{2} \)
97 \( 1 - 6.46iT - 97T^{2} \)
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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

−9.064029520435329919990114478933, −8.636527387524489655812696924863, −7.74937616725416798468188160742, −6.97322530991699450599814182461, −6.12023353479831685382519033227, −5.16246076303349295946094244446, −4.60081516833154000147989296224, −3.88541061467036019778988857693, −1.91037931692925213351977569495, −0.72858526609566145113387874499, 1.00709186437094583752640874075, 2.57443010512052862771545291996, 3.16350334136744636485452345626, 3.79084318576535453955210906054, 5.32253240533825213316405372674, 5.82407778626589457210042884670, 7.29091878261180901313435700972, 7.59094703965422846271846659781, 8.609662482347209776718228317181, 9.638084054611342178042318940532

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