Properties

Degree $2$
Conductor $1764$
Sign $-0.175 + 0.984i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.545 + 1.30i)2-s + (−1.40 − 1.42i)4-s − 0.698i·5-s + (2.62 − 1.05i)8-s + (0.911 + 0.380i)10-s − 2.55·11-s − 1.88·13-s + (−0.0532 + 3.99i)16-s − 3.97i·17-s + 7.05i·19-s + (−0.994 + 0.980i)20-s + (1.39 − 3.32i)22-s + 4.02·23-s + 4.51·25-s + (1.02 − 2.45i)26-s + ⋯
L(s)  = 1  + (−0.385 + 0.922i)2-s + (−0.702 − 0.711i)4-s − 0.312i·5-s + (0.927 − 0.373i)8-s + (0.288 + 0.120i)10-s − 0.769·11-s − 0.521·13-s + (−0.0133 + 0.999i)16-s − 0.963i·17-s + 1.61i·19-s + (−0.222 + 0.219i)20-s + (0.296 − 0.709i)22-s + 0.839·23-s + 0.902·25-s + (0.201 − 0.481i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.175 + 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.175 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.175 + 0.984i$
Motivic weight: \(1\)
Character: $\chi_{1764} (1079, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1764,\ (\ :1/2),\ -0.175 + 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3566573304\)
\(L(\frac12)\) \(\approx\) \(0.3566573304\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.545 - 1.30i)T \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + 0.698iT - 5T^{2} \)
11 \( 1 + 2.55T + 11T^{2} \)
13 \( 1 + 1.88T + 13T^{2} \)
17 \( 1 + 3.97iT - 17T^{2} \)
19 \( 1 - 7.05iT - 19T^{2} \)
23 \( 1 - 4.02T + 23T^{2} \)
29 \( 1 + 1.86iT - 29T^{2} \)
31 \( 1 + 0.941iT - 31T^{2} \)
37 \( 1 + 7.49T + 37T^{2} \)
41 \( 1 + 10.6iT - 41T^{2} \)
43 \( 1 + 3.97iT - 43T^{2} \)
47 \( 1 + 8.90T + 47T^{2} \)
53 \( 1 - 0.529iT - 53T^{2} \)
59 \( 1 + 13.3T + 59T^{2} \)
61 \( 1 + 10.3T + 61T^{2} \)
67 \( 1 + 2.72iT - 67T^{2} \)
71 \( 1 + 3.51T + 71T^{2} \)
73 \( 1 - 2.75T + 73T^{2} \)
79 \( 1 - 13.5iT - 79T^{2} \)
83 \( 1 + 17.1T + 83T^{2} \)
89 \( 1 + 11.8iT - 89T^{2} \)
97 \( 1 + 10.8T + 97T^{2} \)
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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

−8.927885199428749453928275395267, −8.262819620247881066206988476694, −7.45621795635053685971047495082, −6.90369361433062386516723477222, −5.77355009838084710720775723186, −5.19159519471389996852011825522, −4.40937088656905650355956835598, −3.12206896102899205974375061575, −1.64913835470040613686119683269, −0.15651073945978104141489081203, 1.41948091194728482721637279522, 2.70953421739657385612441834866, 3.21492442352150722835261838039, 4.60966510474871628034246564453, 5.07131357531840693749087004126, 6.49093997613668673954293079583, 7.30353771636194456147934893768, 8.107936112052473032060885940930, 8.885486227475516337499885675572, 9.538657094522965517635629161939

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