Properties

Label 2-60e2-5.4-c1-0-20
Degree $2$
Conductor $3600$
Sign $0.894 - 0.447i$
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  + i·7-s + 6·11-s − 5i·13-s + 6i·17-s + 5·19-s + 6i·23-s − 6·29-s + 31-s + 2i·37-s i·43-s + 6i·47-s + 6·49-s − 12i·53-s + 6·59-s − 13·61-s + ⋯
L(s)  = 1  + 0.377i·7-s + 1.80·11-s − 1.38i·13-s + 1.45i·17-s + 1.14·19-s + 1.25i·23-s − 1.11·29-s + 0.179·31-s + 0.328i·37-s − 0.152i·43-s + 0.875i·47-s + 0.857·49-s − 1.64i·53-s + 0.781·59-s − 1.66·61-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(3600\)    =    \(2^{4} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.894 - 0.447i$
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: \((2,\ 3600,\ (\ :1/2),\ 0.894 - 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.167585746\)
\(L(\frac12)\) \(\approx\) \(2.167585746\)
\(L(\frac{3}{2})\) 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 \( 1 - iT - 7T^{2} \)
11 \( 1 - 6T + 11T^{2} \)
13 \( 1 + 5iT - 13T^{2} \)
17 \( 1 - 6iT - 17T^{2} \)
19 \( 1 - 5T + 19T^{2} \)
23 \( 1 - 6iT - 23T^{2} \)
29 \( 1 + 6T + 29T^{2} \)
31 \( 1 - T + 31T^{2} \)
37 \( 1 - 2iT - 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 + iT - 43T^{2} \)
47 \( 1 - 6iT - 47T^{2} \)
53 \( 1 + 12iT - 53T^{2} \)
59 \( 1 - 6T + 59T^{2} \)
61 \( 1 + 13T + 61T^{2} \)
67 \( 1 + 11iT - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 2iT - 73T^{2} \)
79 \( 1 - 8T + 79T^{2} \)
83 \( 1 - 6iT - 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 + 7iT - 97T^{2} \)
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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

−8.629679674318943130088201720546, −7.86179720746328266290098440204, −7.21275665529451835934570381372, −6.17726692128178121108605732325, −5.78113002357881209574159147094, −4.87370293754579118656211810960, −3.65673988935779684183758624800, −3.40283852643378920439273605297, −1.92683348019114953411336751075, −1.04349612918029643659575131225, 0.810328700324261265801925985096, 1.82065939696385397107901668761, 2.97972014926175197134583432130, 4.02582808433159564164356811089, 4.45307051704416614190699098179, 5.50584088868085216328610878521, 6.45244928990761226475218055610, 7.01890935575847896116117761371, 7.51602132357373174389843400926, 8.771812430891067718861478482068

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