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Properties

Label 2-2646-63.20-c1-0-14
Degree $2$
Conductor $2646$
Sign $0.294 - 0.955i$
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  + (0.866 + 0.5i)2-s + (0.499 + 0.866i)4-s + (−0.712 − 1.23i)5-s + 0.999i·8-s − 1.42i·10-s + (2.12 + 1.22i)11-s + (−1.61 + 0.934i)13-s + (−0.5 + 0.866i)16-s − 1.98·17-s + 5.90i·19-s + (0.712 − 1.23i)20-s + (1.22 + 2.12i)22-s + (5.65 − 3.26i)23-s + (1.48 − 2.56i)25-s − 1.86·26-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.249 + 0.433i)4-s + (−0.318 − 0.552i)5-s + 0.353i·8-s − 0.450i·10-s + (0.639 + 0.369i)11-s + (−0.448 + 0.259i)13-s + (−0.125 + 0.216i)16-s − 0.481·17-s + 1.35i·19-s + (0.159 − 0.276i)20-s + (0.261 + 0.452i)22-s + (1.17 − 0.680i)23-s + (0.296 − 0.513i)25-s − 0.366·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.294 - 0.955i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.294 - 0.955i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2646\)    =    \(2 \cdot 3^{3} \cdot 7^{2}\)
Sign: $0.294 - 0.955i$
Analytic conductor: \(21.1284\)
Root analytic conductor: \(4.59656\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2646} (1763, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2646,\ (\ :1/2),\ 0.294 - 0.955i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.342944337\)
\(L(\frac12)\) \(\approx\) \(2.342944337\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 - 0.5i)T \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (0.712 + 1.23i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-2.12 - 1.22i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.61 - 0.934i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + 1.98T + 17T^{2} \)
19 \( 1 - 5.90iT - 19T^{2} \)
23 \( 1 + (-5.65 + 3.26i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-5.51 - 3.18i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (4.15 - 2.39i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 1.83T + 37T^{2} \)
41 \( 1 + (-2.32 - 4.02i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (5.07 - 8.79i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-6.47 + 11.2i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 11.8iT - 53T^{2} \)
59 \( 1 + (2.88 + 5.00i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-8.38 - 4.84i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-7.60 - 13.1i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 0.594iT - 71T^{2} \)
73 \( 1 - 13.5iT - 73T^{2} \)
79 \( 1 + (-4.87 + 8.44i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-1.60 + 2.78i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 0.144T + 89T^{2} \)
97 \( 1 + (2.32 + 1.34i)T + (48.5 + 84.0i)T^{2} \)
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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

−8.700610375758311415799953927087, −8.393858414433905900538864907411, −7.26249449682536181578467267111, −6.78492654879874473884025173973, −5.90999170953172046503843146388, −4.93411640016555340941331820188, −4.42647506321670616722857066843, −3.56274341876798055569565899945, −2.47166714367309436861496885595, −1.20931037628436191595744466688, 0.68752328047774708017138093896, 2.14908842290925766257305470266, 3.06684052227205982619601835355, 3.75608100674250383991641094759, 4.77107227230601910157032932679, 5.41357588125505627434176368505, 6.53813694528250607979844655843, 6.96679861652605375252706897387, 7.78145313415687090439648998620, 8.937867025025159027966635186484

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