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Properties

Label 2-2646-63.4-c1-0-2
Degree $2$
Conductor $2646$
Sign $-0.988 + 0.151i$
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 − 1.37·5-s + 0.999·8-s + (0.686 − 1.18i)10-s − 4.37·11-s + (1 − 1.73i)13-s + (−0.5 + 0.866i)16-s + (2.18 − 3.78i)17-s + (2.5 + 4.33i)19-s + (0.686 + 1.18i)20-s + (2.18 − 3.78i)22-s + 7.37·23-s − 3.11·25-s + (0.999 + 1.73i)26-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s − 0.613·5-s + 0.353·8-s + (0.216 − 0.375i)10-s − 1.31·11-s + (0.277 − 0.480i)13-s + (−0.125 + 0.216i)16-s + (0.530 − 0.918i)17-s + (0.573 + 0.993i)19-s + (0.153 + 0.265i)20-s + (0.466 − 0.807i)22-s + 1.53·23-s − 0.623·25-s + (0.196 + 0.339i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.988 + 0.151i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.988 + 0.151i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2646\)    =    \(2 \cdot 3^{3} \cdot 7^{2}\)
Sign: $-0.988 + 0.151i$
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: \((2,\ 2646,\ (\ :1/2),\ -0.988 + 0.151i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2205679898\)
\(L(\frac12)\) \(\approx\) \(0.2205679898\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.37T + 5T^{2} \)
11 \( 1 + 4.37T + 11T^{2} \)
13 \( 1 + (-1 + 1.73i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2.18 + 3.78i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.5 - 4.33i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 7.37T + 23T^{2} \)
29 \( 1 + (-1.37 - 2.37i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1 - 1.73i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (1 + 1.73i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (5.18 - 8.98i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.55 + 7.89i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1.37 + 2.37i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.55 + 6.16i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (7.05 - 12.2i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (7.55 + 13.0i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 10.1T + 71T^{2} \)
73 \( 1 + (2.55 - 4.43i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (6.05 - 10.4i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-2.74 - 4.75i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (1.62 + 2.81i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-4.55 - 7.89i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.073684140267707047390624338360, −8.349680819714822198987508077116, −7.64657677336109995884671063366, −7.29828602054554547070065834327, −6.23126885540661381447017695455, −5.28276044631606838093298756513, −4.91104502472246924977448718336, −3.56697090247335501410285006059, −2.82150446605565505428618394921, −1.24084607795543531909602264880, 0.090289593067936402860951400937, 1.45549595698793833188549772959, 2.70919360551981538874235039769, 3.38250798146879301115156417147, 4.42416136297729393736808409442, 5.13864956846164832792509945448, 6.15744491760836038406482169502, 7.27944090865775516762036080413, 7.72442662880701774210480160200, 8.566244284610334413542794391053

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