Properties

Label 2-4600-5.4-c1-0-17
Degree $2$
Conductor $4600$
Sign $-0.447 - 0.894i$
Analytic cond. $36.7311$
Root an. cond. $6.06062$
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  + 2.56i·3-s − 5.12i·7-s − 3.56·9-s − 4·11-s − 0.561i·13-s + 3.12i·17-s − 4·19-s + 13.1·21-s + i·23-s − 1.43i·27-s + 8.56·29-s + 1.43·31-s − 10.2i·33-s + 7.12i·37-s + 1.43·39-s + ⋯
L(s)  = 1  + 1.47i·3-s − 1.93i·7-s − 1.18·9-s − 1.20·11-s − 0.155i·13-s + 0.757i·17-s − 0.917·19-s + 2.86·21-s + 0.208i·23-s − 0.276i·27-s + 1.58·29-s + 0.258·31-s − 1.78i·33-s + 1.17i·37-s + 0.230·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4600\)    =    \(2^{3} \cdot 5^{2} \cdot 23\)
Sign: $-0.447 - 0.894i$
Analytic conductor: \(36.7311\)
Root analytic conductor: \(6.06062\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4600} (4049, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4600,\ (\ :1/2),\ -0.447 - 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.189732466\)
\(L(\frac12)\) \(\approx\) \(1.189732466\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.56iT - 3T^{2} \)
7 \( 1 + 5.12iT - 7T^{2} \)
11 \( 1 + 4T + 11T^{2} \)
13 \( 1 + 0.561iT - 13T^{2} \)
17 \( 1 - 3.12iT - 17T^{2} \)
19 \( 1 + 4T + 19T^{2} \)
29 \( 1 - 8.56T + 29T^{2} \)
31 \( 1 - 1.43T + 31T^{2} \)
37 \( 1 - 7.12iT - 37T^{2} \)
41 \( 1 - 0.561T + 41T^{2} \)
43 \( 1 + 9.12iT - 43T^{2} \)
47 \( 1 - 3.68iT - 47T^{2} \)
53 \( 1 + 4.24iT - 53T^{2} \)
59 \( 1 - 6.24T + 59T^{2} \)
61 \( 1 - 11.1T + 61T^{2} \)
67 \( 1 + 6.24iT - 67T^{2} \)
71 \( 1 - 3.68T + 71T^{2} \)
73 \( 1 - 16.5iT - 73T^{2} \)
79 \( 1 - 10.2T + 79T^{2} \)
83 \( 1 - 12iT - 83T^{2} \)
89 \( 1 + 10T + 89T^{2} \)
97 \( 1 - 16.2iT - 97T^{2} \)
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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

−8.315429839397590504650329507313, −8.155148841363971539368570617805, −7.05996016968879004686350362639, −6.44185048243806787754948264090, −5.26975898131392061225082857582, −4.75650403919347359224137648261, −3.99905550864693684333654420467, −3.57741243969026835730123204641, −2.49784827683864749348359778673, −0.928492083624753121380671307462, 0.38589458470359794684406099841, 1.81256664893155474710845073819, 2.52392607575170497083776337443, 2.89187067048410169559233977724, 4.58019422884141578344468285036, 5.37246112457987608088043507193, 6.00005267268244567192551839797, 6.59780161375034138546394315210, 7.35961324255671109235785818333, 8.276108440104847537546956703852

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