Properties

Degree $2$
Conductor $1764$
Sign $0.986 + 0.162i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.986 + 0.162i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.986 + 0.162i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.986 + 0.162i$
Motivic weight: \(1\)
Character: $\chi_{1764} (1079, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1764,\ (\ :1/2),\ 0.986 + 0.162i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.552795626\)
\(L(\frac12)\) \(\approx\) \(1.552795626\)
\(L(\frac{3}{2})\) 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

−9.020285461360826196950294630734, −8.594659310103544118847488217593, −7.36449006619183708026870543738, −7.03367861989983422488173306047, −6.10034544024647216887609021841, −5.12401245211148063707715298471, −4.58045956699554498205347436418, −3.57071786012515292207364137591, −2.45442206086270944200145840891, −0.54799028107928627436316676702, 1.31734600330951595961819512338, 2.28260199480671556929414364391, 3.50707732474794604233987938825, 4.03939988362878674181142103788, 5.12986805304008628204968530752, 6.00326924682747856117958821155, 6.70932608539108138098702153882, 7.926042224574181275948476637616, 8.670227786975697402387146882923, 9.537709925796963240048372539407

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