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Properties

Label 2-1800-120.59-c1-0-20
Degree $2$
Conductor $1800$
Sign $-0.909 - 0.416i$
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  + (1.23 + 0.681i)2-s + (1.07 + 1.68i)4-s − 1.41·7-s + (0.175 + 2.82i)8-s + 6.37i·11-s − 3.54·13-s + (−1.75 − 0.964i)14-s + (−1.70 + 3.61i)16-s + 3.92·17-s − 1.27·19-s + (−4.34 + 7.89i)22-s − 6.28i·23-s + (−4.38 − 2.41i)26-s + (−1.51 − 2.38i)28-s − 9.00·29-s + ⋯
L(s)  = 1  + (0.876 + 0.482i)2-s + (0.535 + 0.844i)4-s − 0.534·7-s + (0.0619 + 0.998i)8-s + 1.92i·11-s − 0.982·13-s + (−0.468 − 0.257i)14-s + (−0.426 + 0.904i)16-s + 0.952·17-s − 0.292·19-s + (−0.925 + 1.68i)22-s − 1.31i·23-s + (−0.860 − 0.473i)26-s + (−0.286 − 0.451i)28-s − 1.67·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.909 - 0.416i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.909 - 0.416i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1800\)    =    \(2^{3} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.909 - 0.416i$
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: \((2,\ 1800,\ (\ :1/2),\ -0.909 - 0.416i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.987264277\)
\(L(\frac12)\) \(\approx\) \(1.987264277\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.23 - 0.681i)T \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + 1.41T + 7T^{2} \)
11 \( 1 - 6.37iT - 11T^{2} \)
13 \( 1 + 3.54T + 13T^{2} \)
17 \( 1 - 3.92T + 17T^{2} \)
19 \( 1 + 1.27T + 19T^{2} \)
23 \( 1 + 6.28iT - 23T^{2} \)
29 \( 1 + 9.00T + 29T^{2} \)
31 \( 1 - 3.92iT - 31T^{2} \)
37 \( 1 - 2.51T + 37T^{2} \)
41 \( 1 + 5.27iT - 41T^{2} \)
43 \( 1 - 1.55iT - 43T^{2} \)
47 \( 1 - 9.73iT - 47T^{2} \)
53 \( 1 + 5.55iT - 53T^{2} \)
59 \( 1 + 0.313iT - 59T^{2} \)
61 \( 1 - 12.7iT - 61T^{2} \)
67 \( 1 - 7.00iT - 67T^{2} \)
71 \( 1 - 0.990T + 71T^{2} \)
73 \( 1 - 12.0iT - 73T^{2} \)
79 \( 1 - 8.18iT - 79T^{2} \)
83 \( 1 - 5.02T + 83T^{2} \)
89 \( 1 + 0.386iT - 89T^{2} \)
97 \( 1 + 10.4iT - 97T^{2} \)
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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

−9.721863343216920634764855331721, −8.735802179198715725097828961530, −7.61046936235129592791840410342, −7.24179986961818469219322127184, −6.47069546042084648331514728434, −5.48550486137567360995839425310, −4.70843780380656977948556456994, −4.01066312773433233581062354382, −2.84337372799513264687876247347, −1.97944396161244207176244551181, 0.50475159322141986511200979344, 1.96805134284422021233521612665, 3.29334894454164279273957924578, 3.51273953171881604032655304896, 4.87052641232108804804863247138, 5.73384579158232749244460589815, 6.14557037921532981067047270741, 7.27690034329562839112585673242, 8.014453183887884604583090068624, 9.267989228979345232308439917747

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