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Properties

Label 2-1800-120.59-c1-0-10
Degree $2$
Conductor $1800$
Sign $-0.986 - 0.161i$
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  + (−0.305 + 1.38i)2-s + (−1.81 − 0.844i)4-s + 1.41·7-s + (1.71 − 2.24i)8-s − 0.191i·11-s − 2.63·13-s + (−0.432 + 1.95i)14-s + (2.57 + 3.06i)16-s − 6.20·17-s + 1.52·19-s + (0.264 + 0.0585i)22-s + 5.25i·23-s + (0.806 − 3.64i)26-s + (−2.56 − 1.19i)28-s + 0.270·29-s + ⋯
L(s)  = 1  + (−0.216 + 0.976i)2-s + (−0.906 − 0.422i)4-s + 0.534·7-s + (0.608 − 0.793i)8-s − 0.0577i·11-s − 0.731·13-s + (−0.115 + 0.521i)14-s + (0.643 + 0.765i)16-s − 1.50·17-s + 0.349·19-s + (0.0563 + 0.0124i)22-s + 1.09i·23-s + (0.158 − 0.714i)26-s + (−0.484 − 0.225i)28-s + 0.0502·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.986 - 0.161i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.986 - 0.161i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1800\)    =    \(2^{3} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.986 - 0.161i$
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: \((2,\ 1800,\ (\ :1/2),\ -0.986 - 0.161i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7841366330\)
\(L(\frac12)\) \(\approx\) \(0.7841366330\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.305 - 1.38i)T \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 - 1.41T + 7T^{2} \)
11 \( 1 + 0.191iT - 11T^{2} \)
13 \( 1 + 2.63T + 13T^{2} \)
17 \( 1 + 6.20T + 17T^{2} \)
19 \( 1 - 1.52T + 19T^{2} \)
23 \( 1 - 5.25iT - 23T^{2} \)
29 \( 1 - 0.270T + 29T^{2} \)
31 \( 1 + 6.20iT - 31T^{2} \)
37 \( 1 - 7.61T + 37T^{2} \)
41 \( 1 - 9.22iT - 41T^{2} \)
43 \( 1 - 12.7iT - 43T^{2} \)
47 \( 1 - 3.79iT - 47T^{2} \)
53 \( 1 - 8.77iT - 53T^{2} \)
59 \( 1 - 10.4iT - 59T^{2} \)
61 \( 1 - 0.382iT - 61T^{2} \)
67 \( 1 - 1.72iT - 67T^{2} \)
71 \( 1 + 9.72T + 71T^{2} \)
73 \( 1 - 5.45iT - 73T^{2} \)
79 \( 1 + 14.3iT - 79T^{2} \)
83 \( 1 + 15.2T + 83T^{2} \)
89 \( 1 + 3.56iT - 89T^{2} \)
97 \( 1 - 7.31iT - 97T^{2} \)
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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

−9.459915756807364471589497414472, −8.835298261119143341159744945612, −7.80171030154586025550428764114, −7.52091288009523318423433688991, −6.42834730504442867607781836906, −5.80834422257077528195073101365, −4.70399026649660941986417513008, −4.31223522956249264323876968419, −2.78644162348145720784561600272, −1.35198472460579489773426299201, 0.32589137984153901536307803869, 1.86547514806560963527879202624, 2.60689866786315935657479257398, 3.81239832401330878526885588187, 4.67136940350032587968625856424, 5.28767247463734351848031532192, 6.65476597303545446364703240875, 7.45244147173704123677610316955, 8.466447033185511372216634390868, 8.860070534575789388555336588098

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