Properties

Label 2-2268-21.20-c3-0-18
Degree $2$
Conductor $2268$
Sign $-0.977 - 0.209i$
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  + 15.6·5-s + (−3.87 + 18.1i)7-s + 39.5i·11-s + 64.1i·13-s − 56.6·17-s − 117. i·19-s + 7.61i·23-s + 119.·25-s + 45.9i·29-s + 290. i·31-s + (−60.6 + 283. i)35-s − 335.·37-s + 194.·41-s − 304.·43-s + 636.·47-s + ⋯
L(s)  = 1  + 1.39·5-s + (−0.209 + 0.977i)7-s + 1.08i·11-s + 1.36i·13-s − 0.808·17-s − 1.41i·19-s + 0.0690i·23-s + 0.959·25-s + 0.294i·29-s + 1.68i·31-s + (−0.293 + 1.36i)35-s − 1.49·37-s + 0.741·41-s − 1.08·43-s + 1.97·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.977 - 0.209i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.977 - 0.209i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2268\)    =    \(2^{2} \cdot 3^{4} \cdot 7\)
Sign: $-0.977 - 0.209i$
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: \((2,\ 2268,\ (\ :3/2),\ -0.977 - 0.209i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.514181528\)
\(L(\frac12)\) \(\approx\) \(1.514181528\)
\(L(\frac{5}{2})\) 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 \( 1 + (3.87 - 18.1i)T \)
good5 \( 1 - 15.6T + 125T^{2} \)
11 \( 1 - 39.5iT - 1.33e3T^{2} \)
13 \( 1 - 64.1iT - 2.19e3T^{2} \)
17 \( 1 + 56.6T + 4.91e3T^{2} \)
19 \( 1 + 117. iT - 6.85e3T^{2} \)
23 \( 1 - 7.61iT - 1.21e4T^{2} \)
29 \( 1 - 45.9iT - 2.43e4T^{2} \)
31 \( 1 - 290. iT - 2.97e4T^{2} \)
37 \( 1 + 335.T + 5.06e4T^{2} \)
41 \( 1 - 194.T + 6.89e4T^{2} \)
43 \( 1 + 304.T + 7.95e4T^{2} \)
47 \( 1 - 636.T + 1.03e5T^{2} \)
53 \( 1 - 274. iT - 1.48e5T^{2} \)
59 \( 1 + 517.T + 2.05e5T^{2} \)
61 \( 1 + 164. iT - 2.26e5T^{2} \)
67 \( 1 + 736.T + 3.00e5T^{2} \)
71 \( 1 + 599. iT - 3.57e5T^{2} \)
73 \( 1 + 214. iT - 3.89e5T^{2} \)
79 \( 1 - 908.T + 4.93e5T^{2} \)
83 \( 1 + 779.T + 5.71e5T^{2} \)
89 \( 1 - 443.T + 7.04e5T^{2} \)
97 \( 1 + 389. iT - 9.12e5T^{2} \)
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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

−9.140348292267742884860920186304, −8.675979491766768965312293491032, −7.11664192869013917971645340078, −6.75330453587357196209692305891, −5.94662441121380899208588816288, −5.04614956005410667654575003115, −4.49831150909250941435918765272, −2.99055412022196330566020746328, −2.12300922351124397398226324156, −1.62469560692141885046712828894, 0.26871314066737299394917775027, 1.25190480508441058694520256034, 2.33048190701869072382021897894, 3.32393466858011966534870293583, 4.17105383506071514088047392039, 5.45612119124800813173908499440, 5.86864627663871227826177756103, 6.57685587199660788334974237032, 7.61337219925168582224731108604, 8.281923937665428985924387633722

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