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Properties

Label 2-2646-63.4-c1-0-26
Degree $2$
Conductor $2646$
Sign $-0.678 + 0.734i$
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 − 3·5-s − 0.999·8-s + (−1.5 + 2.59i)10-s + 3·11-s + (2.5 − 4.33i)13-s + (−0.5 + 0.866i)16-s + (−1.5 + 2.59i)17-s + (2.5 + 4.33i)19-s + (1.49 + 2.59i)20-s + (1.5 − 2.59i)22-s + 3·23-s + 4·25-s + (−2.5 − 4.33i)26-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s − 1.34·5-s − 0.353·8-s + (−0.474 + 0.821i)10-s + 0.904·11-s + (0.693 − 1.20i)13-s + (−0.125 + 0.216i)16-s + (−0.363 + 0.630i)17-s + (0.573 + 0.993i)19-s + (0.335 + 0.580i)20-s + (0.319 − 0.553i)22-s + 0.625·23-s + 0.800·25-s + (−0.490 − 0.849i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.678 + 0.734i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.678 + 0.734i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2646\)    =    \(2 \cdot 3^{3} \cdot 7^{2}\)
Sign: $-0.678 + 0.734i$
Analytic conductor: \(21.1284\)
Root analytic conductor: \(4.59656\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2646} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2646,\ (\ :1/2),\ -0.678 + 0.734i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.350025938\)
\(L(\frac12)\) \(\approx\) \(1.350025938\)
\(L(\frac{3}{2})\) 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 \( 1 + 3T + 5T^{2} \)
11 \( 1 - 3T + 11T^{2} \)
13 \( 1 + (-2.5 + 4.33i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (1.5 - 2.59i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.5 - 4.33i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 3T + 23T^{2} \)
29 \( 1 + (1.5 + 2.59i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (2 + 3.46i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-3.5 - 6.06i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-4.5 + 7.79i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (5.5 + 9.52i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.5 - 2.59i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (6 + 10.3i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1 + 1.73i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2 - 3.46i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + (-5.5 + 9.52i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (4 - 6.92i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (1.5 + 2.59i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (7.5 + 12.9i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (0.5 + 0.866i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.466261785404403585366789354703, −7.982863134860069762339449149595, −7.12289231534284613602805681129, −6.13622529809761780623611474799, −5.39996888839655529982530980354, −4.26704044848022488895253474557, −3.73500277979377280086059963313, −3.11134340122041873026647462203, −1.64391937376548557638798864871, −0.46642438152612757018572555655, 1.15296223501929745547087845591, 2.85085217907666056857592669651, 3.76972021160970352689687162476, 4.37091637727000463277538109781, 5.07242371475270528210500785488, 6.30812179500155645160111115047, 6.89571198473592990235967409170, 7.45630031383933194224886294360, 8.284724291360690195773354397034, 9.127512573061305029425760939494

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