Properties

Label 2-1800-120.59-c1-0-1
Degree $2$
Conductor $1800$
Sign $-0.883 + 0.468i$
Analytic cond. $14.3730$
Root an. cond. $3.79118$
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  + (−0.933 + 1.06i)2-s + (−0.257 − 1.98i)4-s + 1.41·7-s + (2.34 + 1.57i)8-s − 2.31i·11-s − 5.14·13-s + (−1.32 + 1.50i)14-s + (−3.86 + 1.02i)16-s + 5.10·17-s − 8.24·19-s + (2.46 + 2.16i)22-s + 0.969i·23-s + (4.80 − 5.46i)26-s + (−0.364 − 2.80i)28-s − 3.28·29-s + ⋯
L(s)  = 1  + (−0.660 + 0.751i)2-s + (−0.128 − 0.991i)4-s + 0.534·7-s + (0.830 + 0.557i)8-s − 0.699i·11-s − 1.42·13-s + (−0.352 + 0.401i)14-s + (−0.966 + 0.255i)16-s + 1.23·17-s − 1.89·19-s + (0.525 + 0.461i)22-s + 0.202i·23-s + (0.942 − 1.07i)26-s + (−0.0688 − 0.530i)28-s − 0.609·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.883 + 0.468i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.883 + 0.468i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1800\)    =    \(2^{3} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.883 + 0.468i$
Analytic conductor: \(14.3730\)
Root analytic conductor: \(3.79118\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1800} (899, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1800,\ (\ :1/2),\ -0.883 + 0.468i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.03622284584\)
\(L(\frac12)\) \(\approx\) \(0.03622284584\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.933 - 1.06i)T \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 - 1.41T + 7T^{2} \)
11 \( 1 + 2.31iT - 11T^{2} \)
13 \( 1 + 5.14T + 13T^{2} \)
17 \( 1 - 5.10T + 17T^{2} \)
19 \( 1 + 8.24T + 19T^{2} \)
23 \( 1 - 0.969iT - 23T^{2} \)
29 \( 1 + 3.28T + 29T^{2} \)
31 \( 1 + 5.10iT - 31T^{2} \)
37 \( 1 + 3.69T + 37T^{2} \)
41 \( 1 - 4.59iT - 41T^{2} \)
43 \( 1 - 3.21iT - 43T^{2} \)
47 \( 1 - 9.52iT - 47T^{2} \)
53 \( 1 - 7.21iT - 53T^{2} \)
59 \( 1 - 0.862iT - 59T^{2} \)
61 \( 1 - 4.63iT - 61T^{2} \)
67 \( 1 + 5.28iT - 67T^{2} \)
71 \( 1 + 13.2T + 71T^{2} \)
73 \( 1 + 12.5iT - 73T^{2} \)
79 \( 1 - 8.01iT - 79T^{2} \)
83 \( 1 - 7.38T + 83T^{2} \)
89 \( 1 + 10.2iT - 89T^{2} \)
97 \( 1 - 15.7iT - 97T^{2} \)
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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

−9.560353293810416883667126327567, −8.910613243983949298914552206256, −7.86857683527136076526492763663, −7.73269256013784819065188062883, −6.57007854232580796917397124043, −5.85734255559188187782447833481, −5.02169347955094505452566631514, −4.20509931495949043771560964099, −2.67882799991419381966679757182, −1.50563287006318103323872278692, 0.01629837022981757070587415100, 1.69955053919228382603567415933, 2.42975681331743596777432420976, 3.63718514659945051589074711269, 4.56916342232156248071337237222, 5.32880506181145842871288942771, 6.80627548183405827379043659150, 7.36139698100345361919575242766, 8.219460284293052447465942344325, 8.809354128711462161679522651029

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