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Properties

Label 2-60e2-5.4-c1-0-17
Degree $2$
Conductor $3600$
Sign $0.447 - 0.894i$
Analytic cond. $28.7461$
Root an. cond. $5.36154$
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  + 4i·7-s + 4·11-s + 2i·13-s − 2i·17-s + 4·19-s − 4i·23-s − 2·29-s + 8·31-s + 6i·37-s + 6·41-s − 8i·43-s + 4i·47-s − 9·49-s + 6i·53-s + 4·59-s + ⋯
L(s)  = 1  + 1.51i·7-s + 1.20·11-s + 0.554i·13-s − 0.485i·17-s + 0.917·19-s − 0.834i·23-s − 0.371·29-s + 1.43·31-s + 0.986i·37-s + 0.937·41-s − 1.21i·43-s + 0.583i·47-s − 1.28·49-s + 0.824i·53-s + 0.520·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3600\)    =    \(2^{4} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.447 - 0.894i$
Analytic conductor: \(28.7461\)
Root analytic conductor: \(5.36154\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3600} (2449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3600,\ (\ :1/2),\ 0.447 - 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.085116964\)
\(L(\frac12)\) \(\approx\) \(2.085116964\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 - 4iT - 7T^{2} \)
11 \( 1 - 4T + 11T^{2} \)
13 \( 1 - 2iT - 13T^{2} \)
17 \( 1 + 2iT - 17T^{2} \)
19 \( 1 - 4T + 19T^{2} \)
23 \( 1 + 4iT - 23T^{2} \)
29 \( 1 + 2T + 29T^{2} \)
31 \( 1 - 8T + 31T^{2} \)
37 \( 1 - 6iT - 37T^{2} \)
41 \( 1 - 6T + 41T^{2} \)
43 \( 1 + 8iT - 43T^{2} \)
47 \( 1 - 4iT - 47T^{2} \)
53 \( 1 - 6iT - 53T^{2} \)
59 \( 1 - 4T + 59T^{2} \)
61 \( 1 + 2T + 61T^{2} \)
67 \( 1 + 8iT - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 6iT - 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 - 16iT - 83T^{2} \)
89 \( 1 + 6T + 89T^{2} \)
97 \( 1 + 14iT - 97T^{2} \)
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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

−8.756690182490073496571882971275, −8.119898784655666189405058332104, −7.07314686727111333186257507816, −6.41115746029184021664929429370, −5.75677728092029600671672255477, −4.91982095288471812460619357280, −4.11306142573944759136807325809, −3.01827823657281266957111878357, −2.28168465702306408134097253236, −1.13116322332217725516239238654, 0.74834512719839418806093265630, 1.54488994964169664253341407905, 3.04063985223338772958822264983, 3.84799316883910091310946859413, 4.37265774165348561885646622034, 5.43807987370460199476076187177, 6.28478168718728176669131805769, 7.02526217526205429846404035764, 7.61417055690723521146936977285, 8.283218751804618055390832124147

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