Properties

Label 2-2646-21.20-c1-0-43
Degree $2$
Conductor $2646$
Sign $0.654 + 0.755i$
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  + i·2-s − 4-s + 1.73·5-s i·8-s + 1.73i·10-s + 16-s − 3.46·17-s − 6.92i·19-s − 1.73·20-s − 6i·23-s − 2.00·25-s − 9i·29-s + 3.46i·31-s + i·32-s − 3.46i·34-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.5·4-s + 0.774·5-s − 0.353i·8-s + 0.547i·10-s + 0.250·16-s − 0.840·17-s − 1.58i·19-s − 0.387·20-s − 1.25i·23-s − 0.400·25-s − 1.67i·29-s + 0.622i·31-s + 0.176i·32-s − 0.594i·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.654 + 0.755i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.654 + 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2646\)    =    \(2 \cdot 3^{3} \cdot 7^{2}\)
Sign: $0.654 + 0.755i$
Analytic conductor: \(21.1284\)
Root analytic conductor: \(4.59656\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2646} (2645, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2646,\ (\ :1/2),\ 0.654 + 0.755i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.347722683\)
\(L(\frac12)\) \(\approx\) \(1.347722683\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 - 1.73T + 5T^{2} \)
11 \( 1 - 11T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 + 3.46T + 17T^{2} \)
19 \( 1 + 6.92iT - 19T^{2} \)
23 \( 1 + 6iT - 23T^{2} \)
29 \( 1 + 9iT - 29T^{2} \)
31 \( 1 - 3.46iT - 31T^{2} \)
37 \( 1 - 4T + 37T^{2} \)
41 \( 1 - 3.46T + 41T^{2} \)
43 \( 1 + 8T + 43T^{2} \)
47 \( 1 + 3.46T + 47T^{2} \)
53 \( 1 - 3iT - 53T^{2} \)
59 \( 1 + 12.1T + 59T^{2} \)
61 \( 1 + 3.46iT - 61T^{2} \)
67 \( 1 - 14T + 67T^{2} \)
71 \( 1 - 6iT - 71T^{2} \)
73 \( 1 + 12.1iT - 73T^{2} \)
79 \( 1 - 11T + 79T^{2} \)
83 \( 1 + 17.3T + 83T^{2} \)
89 \( 1 + 10.3T + 89T^{2} \)
97 \( 1 + 6.92iT - 97T^{2} \)
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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

−8.696010580154963565569476051719, −8.052469860631481938842447562199, −7.05132751005043851546237952269, −6.45767220955065700803867821336, −5.84785856360911156112255799902, −4.82330606447014498129020397812, −4.31862279008079318451492680646, −2.92452581889107927982024930614, −2.01981427217287942549619939831, −0.43016750187602492803513983418, 1.39985817338617933877792908582, 2.09915777780361480466238075022, 3.24680385189570790314893199631, 4.01788752085104880234932400964, 5.08103161813998801367509538082, 5.76138974144281889441495764637, 6.53766314410365554379487399712, 7.57836009752421170769667389653, 8.333281504971561615490848301153, 9.184247969392342059676951378724

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