Properties

Label 2-2646-63.16-c1-0-17
Degree $2$
Conductor $2646$
Sign $-0.0788 - 0.996i$
Analytic cond. $21.1284$
Root an. cond. $4.59656$
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  + (0.5 + 0.866i)2-s + (−0.499 + 0.866i)4-s − 0.999·8-s + 3·11-s + (1 + 1.73i)13-s + (−0.5 − 0.866i)16-s + (1.5 + 2.59i)17-s + (−0.5 + 0.866i)19-s + (1.5 + 2.59i)22-s + 6·23-s − 5·25-s + (−0.999 + 1.73i)26-s + (3 − 5.19i)29-s + (−2 + 3.46i)31-s + (0.499 − 0.866i)32-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s − 0.353·8-s + 0.904·11-s + (0.277 + 0.480i)13-s + (−0.125 − 0.216i)16-s + (0.363 + 0.630i)17-s + (−0.114 + 0.198i)19-s + (0.319 + 0.553i)22-s + 1.25·23-s − 25-s + (−0.196 + 0.339i)26-s + (0.557 − 0.964i)29-s + (−0.359 + 0.622i)31-s + (0.0883 − 0.153i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0788 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0788 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2646\)    =    \(2 \cdot 3^{3} \cdot 7^{2}\)
Sign: $-0.0788 - 0.996i$
Analytic conductor: \(21.1284\)
Root analytic conductor: \(4.59656\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2646} (667, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2646,\ (\ :1/2),\ -0.0788 - 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.204833154\)
\(L(\frac12)\) \(\approx\) \(2.204833154\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 - 0.866i)T \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + 5T^{2} \)
11 \( 1 - 3T + 11T^{2} \)
13 \( 1 + (-1 - 1.73i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1.5 - 2.59i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 6T + 23T^{2} \)
29 \( 1 + (-3 + 5.19i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (2 - 3.46i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2 + 3.46i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (4.5 + 7.79i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.5 + 0.866i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-3 - 5.19i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-6 - 10.3i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.5 - 2.59i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4 - 6.92i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.5 - 4.33i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 12T + 71T^{2} \)
73 \( 1 + (-5.5 - 9.52i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2 - 3.46i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (6 - 10.3i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (3 - 5.19i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-2.5 + 4.33i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.934572609032773817426846039577, −8.297868267920520617097101934547, −7.38694678498765606700225856026, −6.75496875614406065778576872497, −5.99363409826546343911512309911, −5.31950809387760137887153122025, −4.16514106222346808771996634468, −3.77548366728852827525990345450, −2.50004754535446845544625641126, −1.18747223031186645529145946301, 0.74444858724157812810508037784, 1.84113257286933209092450952087, 3.04244942793068378935527259836, 3.66462531008247542517508268212, 4.70960741806288521832459952492, 5.36060665644419951709524121185, 6.33528618852398443715687928230, 6.99644987445691438954436417764, 8.005442320517725165305069637363, 8.801068843666960516803729481475

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