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Properties

Label 2-1800-8.5-c1-0-3
Degree $2$
Conductor $1800$
Sign $-1$
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  − 1.41·2-s + 2.00·4-s − 2.82·8-s + 4.47i·11-s + 6.32i·13-s + 4.00·16-s − 2.82·17-s − 6.32i·22-s − 5.65·23-s − 8.94i·26-s − 4.47i·29-s + 2·31-s − 5.65·32-s + 4.00·34-s − 6.32i·37-s + ⋯
L(s)  = 1  − 1.00·2-s + 1.00·4-s − 1.00·8-s + 1.34i·11-s + 1.75i·13-s + 1.00·16-s − 0.685·17-s − 1.34i·22-s − 1.17·23-s − 1.75i·26-s − 0.830i·29-s + 0.359·31-s − 1.00·32-s + 0.685·34-s − 1.03i·37-s + ⋯

Functional equation

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

Invariants

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

Particular Values

\(L(1)\) \(\approx\) \(0.2588387047\)
\(L(\frac12)\) \(\approx\) \(0.2588387047\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 1.41T \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + 7T^{2} \)
11 \( 1 - 4.47iT - 11T^{2} \)
13 \( 1 - 6.32iT - 13T^{2} \)
17 \( 1 + 2.82T + 17T^{2} \)
19 \( 1 - 19T^{2} \)
23 \( 1 + 5.65T + 23T^{2} \)
29 \( 1 + 4.47iT - 29T^{2} \)
31 \( 1 - 2T + 31T^{2} \)
37 \( 1 + 6.32iT - 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 + 12.6iT - 43T^{2} \)
47 \( 1 + 11.3T + 47T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + 4.47iT - 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 - 12.6iT - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 73T^{2} \)
79 \( 1 + 14T + 79T^{2} \)
83 \( 1 - 83T^{2} \)
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

−9.699532920774571048251137302249, −8.950267910385861104944017656005, −8.211947507809663605982069080320, −7.23938773462528194519073442107, −6.79528414433338494142087906320, −5.95797769554445600635098437557, −4.66309656866423096409596836507, −3.85517197682056741611312372326, −2.27211372554076332137311828505, −1.78207355844286769919428610011, 0.13086096771456896026793793062, 1.36840164241791941006126396385, 2.82831439352488649218422623630, 3.39395748091631167140877406231, 4.94491453448288443906406302958, 5.99192804942718422061768364657, 6.41224646680758118439370590761, 7.63609072040537044493383988777, 8.213699820038908838266786997457, 8.675579105111793958386751541505

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