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Properties

Label 2-1800-5.4-c1-0-21
Degree $2$
Conductor $1800$
Sign $-0.894 - 0.447i$
Analytic cond. $14.3730$
Root an. cond. $3.79118$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4i·7-s − 4·11-s + 2i·13-s − 2i·17-s − 4·19-s + 4i·23-s − 2·29-s − 8·31-s + 6i·37-s + 6·41-s + 8i·43-s − 4i·47-s − 9·49-s + 6i·53-s − 4·59-s + ⋯
L(s)  = 1  − 1.51i·7-s − 1.20·11-s + 0.554i·13-s − 0.485i·17-s − 0.917·19-s + 0.834i·23-s − 0.371·29-s − 1.43·31-s + 0.986i·37-s + 0.937·41-s + 1.21i·43-s − 0.583i·47-s − 1.28·49-s + 0.824i·53-s − 0.520·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1800\)    =    \(2^{3} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.894 - 0.447i$
Analytic conductor: \(14.3730\)
Root analytic conductor: \(3.79118\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1800} (649, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 1800,\ (\ :1/2),\ -0.894 - 0.447i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + 4iT - 7T^{2} \)
11 \( 1 + 4T + 11T^{2} \)
13 \( 1 - 2iT - 13T^{2} \)
17 \( 1 + 2iT - 17T^{2} \)
19 \( 1 + 4T + 19T^{2} \)
23 \( 1 - 4iT - 23T^{2} \)
29 \( 1 + 2T + 29T^{2} \)
31 \( 1 + 8T + 31T^{2} \)
37 \( 1 - 6iT - 37T^{2} \)
41 \( 1 - 6T + 41T^{2} \)
43 \( 1 - 8iT - 43T^{2} \)
47 \( 1 + 4iT - 47T^{2} \)
53 \( 1 - 6iT - 53T^{2} \)
59 \( 1 + 4T + 59T^{2} \)
61 \( 1 + 2T + 61T^{2} \)
67 \( 1 - 8iT - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 6iT - 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + 16iT - 83T^{2} \)
89 \( 1 + 6T + 89T^{2} \)
97 \( 1 + 14iT - 97T^{2} \)
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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

−8.847709111336608870019396406632, −7.77080944611726142621936812250, −7.41172780847449720363020333335, −6.55968730212721699227884327880, −5.52542560750406947022530639259, −4.58885776614249694861552563657, −3.87672902355966487323261371011, −2.79112454019451606856737674244, −1.48499495784257744217639413273, 0, 2.06277990496082175008236103009, 2.67418249861115509209892080655, 3.85879349896762081795401640517, 5.09528650114999427237624895685, 5.62377263191614303710863542778, 6.36738877198213162555516852131, 7.52119316008319306292488441513, 8.243293411874429110793515009153, 8.873584566447424430706400357654

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