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Properties

Label 2-1800-40.29-c1-0-47
Degree $2$
Conductor $1800$
Sign $0.0534 + 0.998i$
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  + (0.192 − 1.40i)2-s + (−1.92 − 0.540i)4-s − 0.0802i·7-s + (−1.12 + 2.59i)8-s − 2.41i·11-s + 5.26·13-s + (−0.112 − 0.0154i)14-s + (3.41 + 2.08i)16-s − 0.255i·17-s + 6.95i·19-s + (−3.38 − 0.465i)22-s + 1.64i·23-s + (1.01 − 7.38i)26-s + (−0.0433 + 0.154i)28-s − 4.51i·29-s + ⋯
L(s)  = 1  + (0.136 − 0.990i)2-s + (−0.962 − 0.270i)4-s − 0.0303i·7-s + (−0.398 + 0.917i)8-s − 0.728i·11-s + 1.46·13-s + (−0.0300 − 0.00413i)14-s + (0.854 + 0.520i)16-s − 0.0620i·17-s + 1.59i·19-s + (−0.721 − 0.0993i)22-s + 0.343i·23-s + (0.199 − 1.44i)26-s + (−0.00819 + 0.0292i)28-s − 0.838i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0534 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0534 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1800\)    =    \(2^{3} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.0534 + 0.998i$
Analytic conductor: \(14.3730\)
Root analytic conductor: \(3.79118\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1800} (1549, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1800,\ (\ :1/2),\ 0.0534 + 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.732160820\)
\(L(\frac12)\) \(\approx\) \(1.732160820\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.192 + 1.40i)T \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + 0.0802iT - 7T^{2} \)
11 \( 1 + 2.41iT - 11T^{2} \)
13 \( 1 - 5.26T + 13T^{2} \)
17 \( 1 + 0.255iT - 17T^{2} \)
19 \( 1 - 6.95iT - 19T^{2} \)
23 \( 1 - 1.64iT - 23T^{2} \)
29 \( 1 + 4.51iT - 29T^{2} \)
31 \( 1 - 8.29T + 31T^{2} \)
37 \( 1 + 2.67T + 37T^{2} \)
41 \( 1 - 8.11T + 41T^{2} \)
43 \( 1 + 4.08T + 43T^{2} \)
47 \( 1 - 5.70iT - 47T^{2} \)
53 \( 1 + 11.5T + 53T^{2} \)
59 \( 1 + 12.6iT - 59T^{2} \)
61 \( 1 + 11.9iT - 61T^{2} \)
67 \( 1 - 7.27T + 67T^{2} \)
71 \( 1 - 11.3T + 71T^{2} \)
73 \( 1 + 12.0iT - 73T^{2} \)
79 \( 1 + 5.50T + 79T^{2} \)
83 \( 1 - 9.20T + 83T^{2} \)
89 \( 1 - 11.9T + 89T^{2} \)
97 \( 1 + 8.50iT - 97T^{2} \)
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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

−9.286448895511596004048183415048, −8.208049513795729438908050765780, −8.049843548164963710349359632013, −6.30532745964841989626320326164, −5.90603140355384127725380204517, −4.80334305701833238708268369172, −3.78262212866052142211292462225, −3.25795140786956993921843839185, −1.93306158027971086745720629381, −0.859864725954183693724207197434, 0.994835630014687299353259923872, 2.72402558799464435725660059253, 3.87983539133608682012995425726, 4.64963774662861535726847165598, 5.47657025023128544995402318545, 6.45235078406479573041725832925, 6.92142620305589639535761321865, 7.84369179253127366114951097752, 8.688399795072617351670841545349, 9.110260491643527002004439508187

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