Properties

Label 2-1800-8.5-c1-0-91
Degree $2$
Conductor $1800$
Sign $-0.707 - 0.707i$
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 − i)2-s − 2i·4-s − 2·7-s + (−2 − 2i)8-s − 4i·11-s + (−2 + 2i)14-s − 4·16-s − 6·17-s + 4i·19-s + (−4 − 4i)22-s − 4·23-s + 4i·28-s + 6i·29-s + 10·31-s + (−4 + 4i)32-s + ⋯
L(s)  = 1  + (0.707 − 0.707i)2-s i·4-s − 0.755·7-s + (−0.707 − 0.707i)8-s − 1.20i·11-s + (−0.534 + 0.534i)14-s − 16-s − 1.45·17-s + 0.917i·19-s + (−0.852 − 0.852i)22-s − 0.834·23-s + 0.755i·28-s + 1.11i·29-s + 1.79·31-s + (−0.707 + 0.707i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1800\)    =    \(2^{3} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.707 - 0.707i$
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: \((2,\ 1800,\ (\ :1/2),\ -0.707 - 0.707i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6679904337\)
\(L(\frac12)\) \(\approx\) \(0.6679904337\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1 + i)T \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + 2T + 7T^{2} \)
11 \( 1 + 4iT - 11T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 + 6T + 17T^{2} \)
19 \( 1 - 4iT - 19T^{2} \)
23 \( 1 + 4T + 23T^{2} \)
29 \( 1 - 6iT - 29T^{2} \)
31 \( 1 - 10T + 31T^{2} \)
37 \( 1 + 4iT - 37T^{2} \)
41 \( 1 + 10T + 41T^{2} \)
43 \( 1 - 4iT - 43T^{2} \)
47 \( 1 + 4T + 47T^{2} \)
53 \( 1 + 10iT - 53T^{2} \)
59 \( 1 + 8iT - 59T^{2} \)
61 \( 1 + 8iT - 61T^{2} \)
67 \( 1 - 12iT - 67T^{2} \)
71 \( 1 - 4T + 71T^{2} \)
73 \( 1 + 10T + 73T^{2} \)
79 \( 1 + 14T + 79T^{2} \)
83 \( 1 - 83T^{2} \)
89 \( 1 + 14T + 89T^{2} \)
97 \( 1 - 10T + 97T^{2} \)
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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

−8.831032880567445837712472912468, −8.249431749506320013405460716063, −6.75634869648055366208379328080, −6.33035114290747587140794483536, −5.49135307478256597520897035521, −4.50311049206574359300872126957, −3.58007478127175611038371481458, −2.90187725485257266717297813458, −1.69654807277499282291602406887, −0.17848268403632496330964072075, 2.19483216598064543239036093092, 3.05537090306069614944728035551, 4.36661479518166264265115554616, 4.62510904831370408658515021086, 5.90587953654790420190448298493, 6.63203457664901773289278052594, 7.09353897325785887548341595208, 8.085107724007899737535770948726, 8.840673059193597144670569822022, 9.671632871743074861267639454371

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