Properties

Label 2-2300-5.4-c1-0-20
Degree $2$
Conductor $2300$
Sign $0.894 + 0.447i$
Analytic cond. $18.3655$
Root an. cond. $4.28550$
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  i·7-s + 3·9-s + 6·11-s − 6i·13-s + 7i·17-s − 2·19-s + i·23-s + 5·29-s + 31-s − 5i·37-s − 7·41-s − 8i·43-s + 8i·47-s + 6·49-s − 3i·53-s + ⋯
L(s)  = 1  − 0.377i·7-s + 9-s + 1.80·11-s − 1.66i·13-s + 1.69i·17-s − 0.458·19-s + 0.208i·23-s + 0.928·29-s + 0.179·31-s − 0.821i·37-s − 1.09·41-s − 1.21i·43-s + 1.16i·47-s + 0.857·49-s − 0.412i·53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2300\)    =    \(2^{2} \cdot 5^{2} \cdot 23\)
Sign: $0.894 + 0.447i$
Analytic conductor: \(18.3655\)
Root analytic conductor: \(4.28550\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2300} (1749, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2300,\ (\ :1/2),\ 0.894 + 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.192630151\)
\(L(\frac12)\) \(\approx\) \(2.192630151\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
23 \( 1 - iT \)
good3 \( 1 - 3T^{2} \)
7 \( 1 + iT - 7T^{2} \)
11 \( 1 - 6T + 11T^{2} \)
13 \( 1 + 6iT - 13T^{2} \)
17 \( 1 - 7iT - 17T^{2} \)
19 \( 1 + 2T + 19T^{2} \)
29 \( 1 - 5T + 29T^{2} \)
31 \( 1 - T + 31T^{2} \)
37 \( 1 + 5iT - 37T^{2} \)
41 \( 1 + 7T + 41T^{2} \)
43 \( 1 + 8iT - 43T^{2} \)
47 \( 1 - 8iT - 47T^{2} \)
53 \( 1 + 3iT - 53T^{2} \)
59 \( 1 + 13T + 59T^{2} \)
61 \( 1 + 8T + 61T^{2} \)
67 \( 1 + 9iT - 67T^{2} \)
71 \( 1 - 7T + 71T^{2} \)
73 \( 1 - 2iT - 73T^{2} \)
79 \( 1 - 12T + 79T^{2} \)
83 \( 1 - 5iT - 83T^{2} \)
89 \( 1 - 12T + 89T^{2} \)
97 \( 1 - 2iT - 97T^{2} \)
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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

−8.938667363762585019139301567959, −8.156328092982725537936892402085, −7.44603973681503725946351077598, −6.51283795569457881604670024255, −6.05088304034239224982757521415, −4.84717135309007106600490248413, −3.95798706527925428401905688710, −3.43824857914363925674989405022, −1.86147466988428088294065180594, −0.945146199103864715787955007581, 1.16729723209106116756486000253, 2.09251379558286140916139089384, 3.37611267518398884253952123913, 4.43193840923078154212539765016, 4.74061882018660392885311021756, 6.25553465764323785701986718471, 6.71998897891180712927148343543, 7.27660868738329037422494157338, 8.479385405703029372376348648787, 9.277260456880027484865366352228

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