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Properties

Label 2-60e2-20.3-c1-0-11
Degree $2$
Conductor $3600$
Sign $-0.525 - 0.850i$
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  + (2.82 + 2.82i)7-s + 5.65i·11-s + (−3 − 3i)13-s + (−1 + i)17-s + 5.65·19-s + (−2.82 + 2.82i)23-s + 4i·29-s + (−5 + 5i)37-s + (2.82 − 2.82i)43-s + (−2.82 − 2.82i)47-s + 9.00i·49-s + (1 + i)53-s − 11.3·59-s + 4·61-s + (−2.82 − 2.82i)67-s + ⋯
L(s)  = 1  + (1.06 + 1.06i)7-s + 1.70i·11-s + (−0.832 − 0.832i)13-s + (−0.242 + 0.242i)17-s + 1.29·19-s + (−0.589 + 0.589i)23-s + 0.742i·29-s + (−0.821 + 0.821i)37-s + (0.431 − 0.431i)43-s + (−0.412 − 0.412i)47-s + 1.28i·49-s + (0.137 + 0.137i)53-s − 1.47·59-s + 0.512·61-s + (−0.345 − 0.345i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.525 - 0.850i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.525 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3600\)    =    \(2^{4} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.525 - 0.850i$
Analytic conductor: \(28.7461\)
Root analytic conductor: \(5.36154\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3600} (2143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3600,\ (\ :1/2),\ -0.525 - 0.850i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.597827868\)
\(L(\frac12)\) \(\approx\) \(1.597827868\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.82 - 2.82i)T + 7iT^{2} \)
11 \( 1 - 5.65iT - 11T^{2} \)
13 \( 1 + (3 + 3i)T + 13iT^{2} \)
17 \( 1 + (1 - i)T - 17iT^{2} \)
19 \( 1 - 5.65T + 19T^{2} \)
23 \( 1 + (2.82 - 2.82i)T - 23iT^{2} \)
29 \( 1 - 4iT - 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 + (5 - 5i)T - 37iT^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 + (-2.82 + 2.82i)T - 43iT^{2} \)
47 \( 1 + (2.82 + 2.82i)T + 47iT^{2} \)
53 \( 1 + (-1 - i)T + 53iT^{2} \)
59 \( 1 + 11.3T + 59T^{2} \)
61 \( 1 - 4T + 61T^{2} \)
67 \( 1 + (2.82 + 2.82i)T + 67iT^{2} \)
71 \( 1 + 5.65iT - 71T^{2} \)
73 \( 1 + (-3 - 3i)T + 73iT^{2} \)
79 \( 1 - 5.65T + 79T^{2} \)
83 \( 1 + (2.82 - 2.82i)T - 83iT^{2} \)
89 \( 1 - 8iT - 89T^{2} \)
97 \( 1 + (-3 + 3i)T - 97iT^{2} \)
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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

−8.794796025728683514956336449650, −7.913047109868366266505926922030, −7.51390359981155042970980433307, −6.68655004300170128442774677577, −5.45738358289832092741758954516, −5.17197427931218091411101764151, −4.41517846647434849297529089030, −3.18315848910153165069586832434, −2.22214724967517208015401005097, −1.52282754867929191619145003511, 0.46744966646273437289178753980, 1.51035261332147738061972477188, 2.69443330031777719240047533335, 3.71648414277240351807893849025, 4.44931284907403082260903561480, 5.20762519778478018989846180889, 6.04255652033969541337234493063, 6.94392528987117401848515753958, 7.64257309698909325788279014951, 8.167728715279014914564746515498

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