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Properties

Label 2-2646-63.58-c1-0-29
Degree $2$
Conductor $2646$
Sign $0.888 + 0.458i$
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  + 2-s + 4-s + (1 + 1.73i)5-s + 8-s + (1 + 1.73i)10-s + (0.5 − 0.866i)11-s + (3 − 5.19i)13-s + 16-s + (−2.5 − 4.33i)17-s + (3.5 − 6.06i)19-s + (1 + 1.73i)20-s + (0.5 − 0.866i)22-s + (2 + 3.46i)23-s + (0.500 − 0.866i)25-s + (3 − 5.19i)26-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s + (0.447 + 0.774i)5-s + 0.353·8-s + (0.316 + 0.547i)10-s + (0.150 − 0.261i)11-s + (0.832 − 1.44i)13-s + 0.250·16-s + (−0.606 − 1.05i)17-s + (0.802 − 1.39i)19-s + (0.223 + 0.387i)20-s + (0.106 − 0.184i)22-s + (0.417 + 0.722i)23-s + (0.100 − 0.173i)25-s + (0.588 − 1.01i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.888 + 0.458i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.888 + 0.458i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2646\)    =    \(2 \cdot 3^{3} \cdot 7^{2}\)
Sign: $0.888 + 0.458i$
Analytic conductor: \(21.1284\)
Root analytic conductor: \(4.59656\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2646} (1549, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2646,\ (\ :1/2),\ 0.888 + 0.458i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.259559663\)
\(L(\frac12)\) \(\approx\) \(3.259559663\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (-1 - 1.73i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.5 + 0.866i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-3 + 5.19i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (2.5 + 4.33i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.5 + 6.06i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2 - 3.46i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (2 + 3.46i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 6T + 31T^{2} \)
37 \( 1 + (1 - 1.73i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-1.5 + 2.59i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.5 - 0.866i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + (-6 - 10.3i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 7T + 59T^{2} \)
61 \( 1 + 12T + 61T^{2} \)
67 \( 1 - 13T + 67T^{2} \)
71 \( 1 - 8T + 71T^{2} \)
73 \( 1 + (0.5 + 0.866i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + 6T + 79T^{2} \)
83 \( 1 + (-8 - 13.8i)T + (-41.5 + 71.8i)T^{2} \)
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

−8.894016847431953400980822889631, −7.80115269011844376530276373581, −7.13654956025678900262641677957, −6.46198629453036784790016244634, −5.58957686407212049532789372439, −5.09183853714789162143657185009, −3.87245379280636323075064588547, −3.02792937841832216522233550271, −2.45039758087918415575607234136, −0.885080629689844153505684757162, 1.43025321675836320593810492964, 1.99728705025649764464269649188, 3.54758530838991231870481696749, 4.10036243895398022835611411951, 5.02337073875204599186372971155, 5.74800514234666326904050439052, 6.48426227974975914469126083330, 7.18847173992713337705876950338, 8.266326861712310932284682929211, 8.917983940601465354939741345583

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