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Properties

Label 2-2646-9.7-c1-0-40
Degree $2$
Conductor $2646$
Sign $0.173 - 0.984i$
Analytic cond. $21.1284$
Root an. cond. $4.59656$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 − 0.866i)2-s + (−0.499 − 0.866i)4-s + (−1.5 − 2.59i)5-s − 0.999·8-s − 3·10-s + (−1.5 + 2.59i)11-s + (−2.5 − 4.33i)13-s + (−0.5 + 0.866i)16-s − 3·17-s + 5·19-s + (−1.50 + 2.59i)20-s + (1.5 + 2.59i)22-s + (−1.5 − 2.59i)23-s + (−2 + 3.46i)25-s − 5·26-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.670 − 1.16i)5-s − 0.353·8-s − 0.948·10-s + (−0.452 + 0.783i)11-s + (−0.693 − 1.20i)13-s + (−0.125 + 0.216i)16-s − 0.727·17-s + 1.14·19-s + (−0.335 + 0.580i)20-s + (0.319 + 0.553i)22-s + (−0.312 − 0.541i)23-s + (−0.400 + 0.692i)25-s − 0.980·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 - 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2646\)    =    \(2 \cdot 3^{3} \cdot 7^{2}\)
Sign: $0.173 - 0.984i$
Analytic conductor: \(21.1284\)
Root analytic conductor: \(4.59656\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2646} (883, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 2646,\ (\ :1/2),\ 0.173 - 0.984i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.5 + 2.59i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (1.5 - 2.59i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.5 + 4.33i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 3T + 17T^{2} \)
19 \( 1 - 5T + 19T^{2} \)
23 \( 1 + (1.5 + 2.59i)T + (-11.5 + 19.9i)T^{2} \)
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 \( 1 + 7T + 37T^{2} \)
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 \( 1 - 3T + 53T^{2} \)
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 \( 1 - 11T + 73T^{2} \)
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 \( 1 + 15T + 89T^{2} \)
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

−8.387901847266357560179966627489, −7.60781679161887113852505493251, −6.84068023300507468327360517764, −5.44530296844230713101902626469, −5.06748554563734133846518196013, −4.35811987507551330906077778436, −3.38710457187711004919979506467, −2.41257483135952809311374190726, −1.17456615676365439721721575973, 0, 2.13498766129058523281645215747, 3.19431810511665199287631834411, 3.79661989205626370787519291274, 4.80750536933182163519769656475, 5.63070909892159303092380506455, 6.66046977742225867813537710065, 6.99784747435667413961537881784, 7.78158769926428687523227948848, 8.441141519039274940130908654574

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