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Properties

Label 2-42e2-84.47-c0-0-3
Degree $2$
Conductor $1764$
Sign $-0.720 - 0.693i$
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  + (0.866 + 0.5i)2-s + (0.499 + 0.866i)4-s + (−0.923 + 1.60i)5-s + 0.999i·8-s + (−1.60 + 0.923i)10-s − 0.765i·13-s + (−0.5 + 0.866i)16-s + (0.382 + 0.662i)17-s − 1.84·20-s + (−1.20 − 2.09i)25-s + (0.382 − 0.662i)26-s + (−0.866 + 0.499i)32-s + 0.765i·34-s + (−0.707 + 1.22i)37-s + (−1.60 − 0.923i)40-s − 0.765·41-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)2-s + (0.499 + 0.866i)4-s + (−0.923 + 1.60i)5-s + 0.999i·8-s + (−1.60 + 0.923i)10-s − 0.765i·13-s + (−0.5 + 0.866i)16-s + (0.382 + 0.662i)17-s − 1.84·20-s + (−1.20 − 2.09i)25-s + (0.382 − 0.662i)26-s + (−0.866 + 0.499i)32-s + 0.765i·34-s + (−0.707 + 1.22i)37-s + (−1.60 − 0.923i)40-s − 0.765·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.720 - 0.693i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.720 - 0.693i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.720 - 0.693i$
Analytic conductor: \(0.880350\)
Root analytic conductor: \(0.938270\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (215, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1764,\ (\ :0),\ -0.720 - 0.693i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.503480020\)
\(L(\frac12)\) \(\approx\) \(1.503480020\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 - 0.5i)T \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (0.923 - 1.60i)T + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + 0.765iT - T^{2} \)
17 \( 1 + (-0.382 - 0.662i)T + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.5 - 0.866i)T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + (0.707 - 1.22i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 + 0.765T + T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (0.5 + 0.866i)T^{2} \)
53 \( 1 + (-1.22 + 0.707i)T + (0.5 - 0.866i)T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (-1.60 - 0.923i)T + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (-1.60 + 0.923i)T + (0.5 - 0.866i)T^{2} \)
79 \( 1 + (0.5 + 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (-0.923 + 1.60i)T + (-0.5 - 0.866i)T^{2} \)
97 \( 1 + 1.84iT - T^{2} \)
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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

−10.17610673070879892125960533014, −8.568887003604623103621677740967, −7.963337988475289814993052690168, −7.26307299380992305988095806196, −6.64167428006915078841579938634, −5.90063704736479115758730900900, −4.87772856695002523657965067371, −3.71294901667583225909262881383, −3.33287072407003277890059011957, −2.32722492502241117114289205460, 0.877955672697376951520529859658, 2.09158838860886240809228948874, 3.55781220729518562522688689317, 4.17280273572742934561475699948, 5.01440499330717945606602644489, 5.51522205598165557853560080906, 6.76865929761692387931830169831, 7.55706841544811138218050440988, 8.529933684671257888618101916460, 9.220420642807406227430504707823

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