Properties

Label 2-60e2-5.4-c1-0-30
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 − 6i·13-s − 2i·17-s + 4·19-s − 8i·23-s − 6·29-s + 6i·37-s − 10·41-s + 4i·43-s − 8i·47-s − 9·49-s − 10i·53-s + 6·61-s − 4i·67-s − 14i·73-s + ⋯
L(s)  = 1  + 1.51i·7-s − 1.66i·13-s − 0.485i·17-s + 0.917·19-s − 1.66i·23-s − 1.11·29-s + 0.986i·37-s − 1.56·41-s + 0.609i·43-s − 1.16i·47-s − 1.28·49-s − 1.37i·53-s + 0.768·61-s − 0.488i·67-s − 1.63i·73-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

\(L(1)\) \(\approx\) \(1.444187553\)
\(L(\frac12)\) \(\approx\) \(1.444187553\)
\(L(\frac{3}{2})\) 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 \( 1 + 11T^{2} \)
13 \( 1 + 6iT - 13T^{2} \)
17 \( 1 + 2iT - 17T^{2} \)
19 \( 1 - 4T + 19T^{2} \)
23 \( 1 + 8iT - 23T^{2} \)
29 \( 1 + 6T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 - 6iT - 37T^{2} \)
41 \( 1 + 10T + 41T^{2} \)
43 \( 1 - 4iT - 43T^{2} \)
47 \( 1 + 8iT - 47T^{2} \)
53 \( 1 + 10iT - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 6T + 61T^{2} \)
67 \( 1 + 4iT - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 14iT - 73T^{2} \)
79 \( 1 - 16T + 79T^{2} \)
83 \( 1 - 12iT - 83T^{2} \)
89 \( 1 - 2T + 89T^{2} \)
97 \( 1 + 2iT - 97T^{2} \)
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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

−8.356117237744262396627248788970, −7.925478168104846975714341066427, −6.86353416069920918335828183447, −6.10896135958146668962288565608, −5.26376993826245499839177109051, −4.99488634805347158017044188817, −3.48334415318213237335190769255, −2.86124222627723501138773372185, −1.98220964954188515704947380266, −0.46173038629256383901340699554, 1.14025669231913413658139872789, 1.97389691906257019638997056139, 3.51830413734142120093643623464, 3.90160661191666830227191516115, 4.77790411426144299987651109889, 5.69531898136205872570559211775, 6.61532021240683204246700409646, 7.37634596758713690071186131137, 7.58645031082790898388984184262, 8.793125230681638335230126352476

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