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Properties

Label 2-1800-8.5-c1-0-49
Degree $2$
Conductor $1800$
Sign $0.221 - 0.975i$
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  + (1.16 + 0.796i)2-s + (0.731 + 1.86i)4-s + 4.72·7-s + (−0.627 + 2.75i)8-s − 3.93i·11-s + 3.46i·13-s + (5.51 + 3.76i)14-s + (−2.93 + 2.72i)16-s − 3.51·17-s + 5.44i·19-s + (3.13 − 4.59i)22-s + 7.11·23-s + (−2.76 + 4.05i)26-s + (3.45 + 8.79i)28-s − 3.66i·29-s + ⋯
L(s)  = 1  + (0.826 + 0.563i)2-s + (0.365 + 0.930i)4-s + 1.78·7-s + (−0.221 + 0.975i)8-s − 1.18i·11-s + 0.961i·13-s + (1.47 + 1.00i)14-s + (−0.732 + 0.680i)16-s − 0.852·17-s + 1.24i·19-s + (0.667 − 0.979i)22-s + 1.48·23-s + (−0.541 + 0.794i)26-s + (0.652 + 1.66i)28-s − 0.681i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.221 - 0.975i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.221 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1800\)    =    \(2^{3} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.221 - 0.975i$
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: \((2,\ 1800,\ (\ :1/2),\ 0.221 - 0.975i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.434893482\)
\(L(\frac12)\) \(\approx\) \(3.434893482\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.16 - 0.796i)T \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 - 4.72T + 7T^{2} \)
11 \( 1 + 3.93iT - 11T^{2} \)
13 \( 1 - 3.46iT - 13T^{2} \)
17 \( 1 + 3.51T + 17T^{2} \)
19 \( 1 - 5.44iT - 19T^{2} \)
23 \( 1 - 7.11T + 23T^{2} \)
29 \( 1 + 3.66iT - 29T^{2} \)
31 \( 1 - 5.23T + 31T^{2} \)
37 \( 1 - 0.414iT - 37T^{2} \)
41 \( 1 + 3.00T + 41T^{2} \)
43 \( 1 + 5.34iT - 43T^{2} \)
47 \( 1 - 0.925T + 47T^{2} \)
53 \( 1 + 0.233iT - 53T^{2} \)
59 \( 1 - 14.3iT - 59T^{2} \)
61 \( 1 - 0.118iT - 61T^{2} \)
67 \( 1 + 13.4iT - 67T^{2} \)
71 \( 1 + 2.19T + 71T^{2} \)
73 \( 1 + 0.563T + 73T^{2} \)
79 \( 1 + 10.2T + 79T^{2} \)
83 \( 1 - 11.3iT - 83T^{2} \)
89 \( 1 + 8.88T + 89T^{2} \)
97 \( 1 - 7.27T + 97T^{2} \)
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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

−8.987356075449137662294473698973, −8.501468370590194205652953736604, −7.85920618027240808081549913168, −7.01523312862494363196867189486, −6.13409577033365578923842539341, −5.33211848170019868540055759584, −4.57692807821606589964671879532, −3.88266718844882848367706692076, −2.62937129410050142870613297897, −1.51336513844007898995466603379, 1.10145356073432137706598020735, 2.12392690817570772892093802394, 3.01180060900408815649492181154, 4.50008575474733545562257901326, 4.75822209445286647008906692799, 5.47170795209503248382745904516, 6.76994634076130472099347415943, 7.33963502184103794434884222573, 8.364366951610917477316715685006, 9.159388675101431118402197865360

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