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Properties

Label 2-1800-8.5-c1-0-24
Degree $2$
Conductor $1800$
Sign $0.918 - 0.395i$
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  + (−0.866 − 1.11i)2-s + (−0.500 + 1.93i)4-s + (2.59 − 1.11i)8-s + (−3.5 − 1.93i)16-s + 6.92·17-s + 7.74i·19-s − 3.46·23-s − 8·31-s + (0.866 + 5.59i)32-s + (−5.99 − 7.74i)34-s + (8.66 − 6.70i)38-s + (2.99 + 3.87i)46-s + 10.3·47-s − 7·49-s − 4.47i·53-s + ⋯
L(s)  = 1  + (−0.612 − 0.790i)2-s + (−0.250 + 0.968i)4-s + (0.918 − 0.395i)8-s + (−0.875 − 0.484i)16-s + 1.68·17-s + 1.77i·19-s − 0.722·23-s − 1.43·31-s + (0.153 + 0.988i)32-s + (−1.02 − 1.32i)34-s + (1.40 − 1.08i)38-s + (0.442 + 0.571i)46-s + 1.51·47-s − 49-s − 0.614i·53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.918 - 0.395i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.918 - 0.395i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1800\)    =    \(2^{3} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.918 - 0.395i$
Analytic conductor: \(14.3730\)
Root analytic conductor: \(3.79118\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1800} (901, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1800,\ (\ :1/2),\ 0.918 - 0.395i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.055321455\)
\(L(\frac12)\) \(\approx\) \(1.055321455\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 + 1.11i)T \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + 7T^{2} \)
11 \( 1 - 11T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 - 6.92T + 17T^{2} \)
19 \( 1 - 7.74iT - 19T^{2} \)
23 \( 1 + 3.46T + 23T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + 8T + 31T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 - 10.3T + 47T^{2} \)
53 \( 1 + 4.47iT - 53T^{2} \)
59 \( 1 - 59T^{2} \)
61 \( 1 - 15.4iT - 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 73T^{2} \)
79 \( 1 - 16T + 79T^{2} \)
83 \( 1 - 17.8iT - 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 + 97T^{2} \)
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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

−9.519430344134181210396996634656, −8.580727197257253203308864321040, −7.84633916331070766490687647079, −7.35717248841602842707719516565, −6.07064781611415331580149160539, −5.25555533083884475872486370678, −3.92913970781789597433695420569, −3.43884554998138423332445990309, −2.16586356876892546660037363489, −1.14797915462694467969125104946, 0.56001091085292731899128418294, 1.91019923050045945912745398064, 3.28649284047911519518160825321, 4.52544734003337993892693007849, 5.36488957655007673724988709798, 6.05936850327190900737971682816, 7.03722384584141532150367763841, 7.59787113433740042300219933220, 8.373642582950540217910674550369, 9.244897399939451695818003998147

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