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Properties

Label 2-2646-21.20-c1-0-49
Degree $2$
Conductor $2646$
Sign $0.755 + 0.654i$
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 + 3.46·5-s − i·8-s + 3.46i·10-s − 5.19i·13-s + 16-s − 6.92·17-s − 3.46i·19-s − 3.46·20-s − 6i·23-s + 6.99·25-s + 5.19·26-s − 8.66i·31-s + i·32-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.5·4-s + 1.54·5-s − 0.353i·8-s + 1.09i·10-s − 1.44i·13-s + 0.250·16-s − 1.68·17-s − 0.794i·19-s − 0.774·20-s − 1.25i·23-s + 1.39·25-s + 1.01·26-s − 1.55i·31-s + 0.176i·32-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(2646\)    =    \(2 \cdot 3^{3} \cdot 7^{2}\)
Sign: $0.755 + 0.654i$
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: \((2,\ 2646,\ (\ :1/2),\ 0.755 + 0.654i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.785495944\)
\(L(\frac12)\) \(\approx\) \(1.785495944\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.46T + 5T^{2} \)
11 \( 1 - 11T^{2} \)
13 \( 1 + 5.19iT - 13T^{2} \)
17 \( 1 + 6.92T + 17T^{2} \)
19 \( 1 + 3.46iT - 19T^{2} \)
23 \( 1 + 6iT - 23T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + 8.66iT - 31T^{2} \)
37 \( 1 + 5T + 37T^{2} \)
41 \( 1 - 6.92T + 41T^{2} \)
43 \( 1 - T + 43T^{2} \)
47 \( 1 + 6.92T + 47T^{2} \)
53 \( 1 + 6iT - 53T^{2} \)
59 \( 1 - 6.92T + 59T^{2} \)
61 \( 1 + 1.73iT - 61T^{2} \)
67 \( 1 + 13T + 67T^{2} \)
71 \( 1 - 6iT - 71T^{2} \)
73 \( 1 - 6.92iT - 73T^{2} \)
79 \( 1 + 7T + 79T^{2} \)
83 \( 1 + 3.46T + 83T^{2} \)
89 \( 1 - 10.3T + 89T^{2} \)
97 \( 1 - 1.73iT - 97T^{2} \)
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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

−8.796777210373156774121994570804, −8.083765109904350309902363750405, −7.05116008232255745317238028411, −6.39700570976664773637345725460, −5.79775956898729590476588660660, −5.06978698900070101025934260169, −4.26914525237756257470380061858, −2.83476177083578911548840288168, −2.10158385709823255651181311462, −0.54385566448260195604611797390, 1.58968893226636122928646063207, 1.95296863211110030335621585762, 3.05749750129489386307356277901, 4.18915930416419034521806935969, 4.95467483256283754615820370696, 5.85401882131939502689448848630, 6.52246994918220913367036466960, 7.30372769927060949444947666784, 8.643011133889375601910259443347, 9.096946106508501109669198974281

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