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Properties

Label 2-1800-8.5-c1-0-88
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

−9.224487259201397904226285831273, −8.228018883605467338854586884519, −6.86034911301390522748682882917, −6.53044525322308556456965770398, −5.19814910301007978763267936044, −4.75114474239013400859558093264, −3.72499021677132885589916709672, −2.74499101876217393679021559289, −1.84056588516667624057872045793, −0.29425142505714949834296604268, 1.91990876317527266858473834235, 3.18447596422298736151619903959, 4.05169193048543115347778490065, 4.86780490686620845028162658418, 5.76838957840655984849898032934, 6.48257260966286189003781153118, 7.24773251447537438121719002651, 8.031971831804253671476416800275, 8.782163323088016888889263595810, 9.444246884475609162091025842247

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