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Properties

Label 2-1800-8.5-c1-0-13
Degree $2$
Conductor $1800$
Sign $0.201 - 0.979i$
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  + (1.26 − 0.622i)2-s + (1.22 − 1.58i)4-s − 3.44·7-s + (0.570 − 2.77i)8-s + 5.54i·11-s + 3.87i·13-s + (−4.38 + 2.14i)14-s + (−0.999 − 3.87i)16-s − 6.22·17-s + 7.03i·19-s + (3.44 + 7.03i)22-s + 1.14·23-s + (2.41 + 4.91i)26-s + (−4.22 + 5.45i)28-s + 3.05i·29-s + ⋯
L(s)  = 1  + (0.897 − 0.440i)2-s + (0.612 − 0.790i)4-s − 1.30·7-s + (0.201 − 0.979i)8-s + 1.67i·11-s + 1.07i·13-s + (−1.17 + 0.573i)14-s + (−0.249 − 0.968i)16-s − 1.50·17-s + 1.61i·19-s + (0.735 + 1.49i)22-s + 0.238·23-s + (0.472 + 0.964i)26-s + (−0.798 + 1.03i)28-s + 0.566i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.201 - 0.979i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.201 - 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1800\)    =    \(2^{3} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.201 - 0.979i$
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: \((2,\ 1800,\ (\ :1/2),\ 0.201 - 0.979i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.513904163\)
\(L(\frac12)\) \(\approx\) \(1.513904163\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.26 + 0.622i)T \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + 3.44T + 7T^{2} \)
11 \( 1 - 5.54iT - 11T^{2} \)
13 \( 1 - 3.87iT - 13T^{2} \)
17 \( 1 + 6.22T + 17T^{2} \)
19 \( 1 - 7.03iT - 19T^{2} \)
23 \( 1 - 1.14T + 23T^{2} \)
29 \( 1 - 3.05iT - 29T^{2} \)
31 \( 1 + 1.44T + 31T^{2} \)
37 \( 1 + 6.32iT - 37T^{2} \)
41 \( 1 + 3.93T + 41T^{2} \)
43 \( 1 - 0.710iT - 43T^{2} \)
47 \( 1 - 10.1T + 47T^{2} \)
53 \( 1 - 8.03iT - 53T^{2} \)
59 \( 1 - 4.98iT - 59T^{2} \)
61 \( 1 + 10.1iT - 61T^{2} \)
67 \( 1 + 5.61iT - 67T^{2} \)
71 \( 1 - 13.5T + 71T^{2} \)
73 \( 1 + 6.89T + 73T^{2} \)
79 \( 1 - 2.89T + 79T^{2} \)
83 \( 1 - 6.10iT - 83T^{2} \)
89 \( 1 + 7.87T + 89T^{2} \)
97 \( 1 + 15.6T + 97T^{2} \)
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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

−9.519606066952016262986526670409, −9.069654008452313505260867722649, −7.51769907768676781278841837709, −6.78239193830795749799612229931, −6.35049561639465549490222002118, −5.27748595289299874386002846065, −4.26751570663512205769063050635, −3.78673269623350533239327895471, −2.50753049521749821033022489193, −1.73101791808793490699513164683, 0.37742151027453818120113416986, 2.63597182264489210143635697662, 3.12713704020129484315146803348, 4.06164152386115811082591442304, 5.15026554015674274430050069480, 5.93598387379478126585284394430, 6.58408021632545488445492597881, 7.19460394342906831141939565500, 8.424574139461906205458282255052, 8.786279626182627650808264959303

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