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Properties

Label 2-1800-120.59-c1-0-29
Degree $2$
Conductor $1800$
Sign $-0.544 - 0.838i$
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  + (0.980 + 1.01i)2-s + (−0.0774 + 1.99i)4-s + 4.53·7-s + (−2.11 + 1.88i)8-s + 5.63i·11-s − 2.34·13-s + (4.44 + 4.61i)14-s + (−3.98 − 0.309i)16-s + 5.85·17-s − 3.92·19-s + (−5.74 + 5.52i)22-s + 0.438i·23-s + (−2.30 − 2.39i)26-s + (−0.351 + 9.05i)28-s + 6.09·29-s + ⋯
L(s)  = 1  + (0.693 + 0.720i)2-s + (−0.0387 + 0.999i)4-s + 1.71·7-s + (−0.746 + 0.664i)8-s + 1.70i·11-s − 0.651·13-s + (1.18 + 1.23i)14-s + (−0.996 − 0.0774i)16-s + 1.42·17-s − 0.901·19-s + (−1.22 + 1.17i)22-s + 0.0913i·23-s + (−0.451 − 0.469i)26-s + (−0.0663 + 1.71i)28-s + 1.13·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.544 - 0.838i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.544 - 0.838i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1800\)    =    \(2^{3} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.544 - 0.838i$
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: \((2,\ 1800,\ (\ :1/2),\ -0.544 - 0.838i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.841349710\)
\(L(\frac12)\) \(\approx\) \(2.841349710\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.980 - 1.01i)T \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 - 4.53T + 7T^{2} \)
11 \( 1 - 5.63iT - 11T^{2} \)
13 \( 1 + 2.34T + 13T^{2} \)
17 \( 1 - 5.85T + 17T^{2} \)
19 \( 1 + 3.92T + 19T^{2} \)
23 \( 1 - 0.438iT - 23T^{2} \)
29 \( 1 - 6.09T + 29T^{2} \)
31 \( 1 + 9.84iT - 31T^{2} \)
37 \( 1 + 6.38T + 37T^{2} \)
41 \( 1 - 4.00iT - 41T^{2} \)
43 \( 1 - 9.90iT - 43T^{2} \)
47 \( 1 - 4.89iT - 47T^{2} \)
53 \( 1 + 1.89iT - 53T^{2} \)
59 \( 1 - 5.11iT - 59T^{2} \)
61 \( 1 + 1.90iT - 61T^{2} \)
67 \( 1 + 0.691iT - 67T^{2} \)
71 \( 1 + 2.18T + 71T^{2} \)
73 \( 1 - 0.429iT - 73T^{2} \)
79 \( 1 - 2.47iT - 79T^{2} \)
83 \( 1 - 2.28T + 83T^{2} \)
89 \( 1 + 8.28iT - 89T^{2} \)
97 \( 1 - 9.09iT - 97T^{2} \)
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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

−9.477294560314360337067330296110, −8.369961338511081525603419456473, −7.77372559981613117993900675340, −7.35399337225270765284541911172, −6.32018627305966307407171400569, −5.29826718150528854839701627063, −4.67476124688195252125319431406, −4.17547816863108577071077370540, −2.67249833656738564336755013088, −1.70087707973551536397919825752, 0.875453258294623026470373156700, 1.90904853848525849640012271636, 3.05647963517788647613453600443, 3.91794724440630026933346538107, 5.08358845850837898263798837213, 5.32062467355478202232905169863, 6.38417966674393226016459727321, 7.44871302657475289346712897113, 8.531287356128863603604292757895, 8.749821897143203243141599761955

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