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Properties

Label 2-2268-21.20-c3-0-65
Degree $2$
Conductor $2268$
Sign $0.0138 + 0.999i$
Analytic cond. $133.816$
Root an. cond. $11.5679$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 18.2·5-s + (18.5 − 0.256i)7-s − 56.9i·11-s − 10.8i·13-s + 65.2·17-s + 36.6i·19-s + 81.1i·23-s + 207.·25-s − 269. i·29-s + 135. i·31-s + (−337. + 4.68i)35-s − 125.·37-s + 235.·41-s − 45.2·43-s + 482.·47-s + ⋯
L(s)  = 1  − 1.63·5-s + (0.999 − 0.0138i)7-s − 1.56i·11-s − 0.230i·13-s + 0.931·17-s + 0.441i·19-s + 0.735i·23-s + 1.66·25-s − 1.72i·29-s + 0.787i·31-s + (−1.63 + 0.0226i)35-s − 0.558·37-s + 0.895·41-s − 0.160·43-s + 1.49·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0138 + 0.999i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.0138 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2268\)    =    \(2^{2} \cdot 3^{4} \cdot 7\)
Sign: $0.0138 + 0.999i$
Analytic conductor: \(133.816\)
Root analytic conductor: \(11.5679\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{2268} (1133, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2268,\ (\ :3/2),\ 0.0138 + 0.999i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.475339127\)
\(L(\frac12)\) \(\approx\) \(1.475339127\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 + (-18.5 + 0.256i)T \)
good5 \( 1 + 18.2T + 125T^{2} \)
11 \( 1 + 56.9iT - 1.33e3T^{2} \)
13 \( 1 + 10.8iT - 2.19e3T^{2} \)
17 \( 1 - 65.2T + 4.91e3T^{2} \)
19 \( 1 - 36.6iT - 6.85e3T^{2} \)
23 \( 1 - 81.1iT - 1.21e4T^{2} \)
29 \( 1 + 269. iT - 2.43e4T^{2} \)
31 \( 1 - 135. iT - 2.97e4T^{2} \)
37 \( 1 + 125.T + 5.06e4T^{2} \)
41 \( 1 - 235.T + 6.89e4T^{2} \)
43 \( 1 + 45.2T + 7.95e4T^{2} \)
47 \( 1 - 482.T + 1.03e5T^{2} \)
53 \( 1 + 70.1iT - 1.48e5T^{2} \)
59 \( 1 + 353.T + 2.05e5T^{2} \)
61 \( 1 - 591. iT - 2.26e5T^{2} \)
67 \( 1 - 523.T + 3.00e5T^{2} \)
71 \( 1 + 895. iT - 3.57e5T^{2} \)
73 \( 1 - 982. iT - 3.89e5T^{2} \)
79 \( 1 + 1.02e3T + 4.93e5T^{2} \)
83 \( 1 + 304.T + 5.71e5T^{2} \)
89 \( 1 + 1.02e3T + 7.04e5T^{2} \)
97 \( 1 + 782. iT - 9.12e5T^{2} \)
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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

−8.315685181457778719027120453177, −7.80309500806801545387959498549, −7.28548057907472151540321018720, −5.97855374947842740895007714504, −5.33998664989243909482209725276, −4.24697293863865482394323620493, −3.67973036085302448575126946183, −2.81605862956791695310748408997, −1.23664301535169425839463222076, −0.40873561191316079628131878084, 0.877685877999120106652754205814, 2.00712471725055080050785814535, 3.19239240292795119000845442539, 4.26151823768130052866108245148, 4.59268349722642044594599487531, 5.51008419216182555837917962918, 6.98025975247083491196451260526, 7.31530511192671868566163371715, 8.008828319102022701426293617592, 8.670052649529091733999256762122

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