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Properties

Label 2-4600-5.4-c1-0-82
Degree $2$
Conductor $4600$
Sign $-0.894 + 0.447i$
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  − 2i·3-s − 3i·7-s − 9-s + 5·11-s + 5i·13-s − 4i·17-s − 19-s − 6·21-s + i·23-s − 4i·27-s − 9·29-s − 2·31-s − 10i·33-s − 2i·37-s + 10·39-s + ⋯
L(s)  = 1  − 1.15i·3-s − 1.13i·7-s − 0.333·9-s + 1.50·11-s + 1.38i·13-s − 0.970i·17-s − 0.229·19-s − 1.30·21-s + 0.208i·23-s − 0.769i·27-s − 1.67·29-s − 0.359·31-s − 1.74i·33-s − 0.328i·37-s + 1.60·39-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(4600\)    =    \(2^{3} \cdot 5^{2} \cdot 23\)
Sign: $-0.894 + 0.447i$
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: \((2,\ 4600,\ (\ :1/2),\ -0.894 + 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.778738296\)
\(L(\frac12)\) \(\approx\) \(1.778738296\)
\(L(\frac{3}{2})\) 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 \( 1 + 2iT - 3T^{2} \)
7 \( 1 + 3iT - 7T^{2} \)
11 \( 1 - 5T + 11T^{2} \)
13 \( 1 - 5iT - 13T^{2} \)
17 \( 1 + 4iT - 17T^{2} \)
19 \( 1 + T + 19T^{2} \)
29 \( 1 + 9T + 29T^{2} \)
31 \( 1 + 2T + 31T^{2} \)
37 \( 1 + 2iT - 37T^{2} \)
41 \( 1 - 3T + 41T^{2} \)
43 \( 1 - 7iT - 43T^{2} \)
47 \( 1 + 12iT - 47T^{2} \)
53 \( 1 + 12iT - 53T^{2} \)
59 \( 1 - 6T + 59T^{2} \)
61 \( 1 + 10T + 61T^{2} \)
67 \( 1 + 8iT - 67T^{2} \)
71 \( 1 - 2T + 71T^{2} \)
73 \( 1 + iT - 73T^{2} \)
79 \( 1 - 11T + 79T^{2} \)
83 \( 1 + 9iT - 83T^{2} \)
89 \( 1 - 14T + 89T^{2} \)
97 \( 1 + 16iT - 97T^{2} \)
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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

−7.63822396663609453925233187010, −7.27927731261662157156767611314, −6.67033661628611123229130692787, −6.24038310374922963070809859646, −4.99545581530395674895248766149, −4.08000153979078475101532858048, −3.60907052529555547302216594091, −2.09013074114321655717971329401, −1.51594355725472460931701144108, −0.50674597202377969495523933350, 1.33932896517752793247981020920, 2.49694239142338475435054588618, 3.53633131033904080334257543496, 3.99802892059942315797710506709, 4.91478241195045360768129282662, 5.76136297393160432215938394070, 6.09450216737845948746616268209, 7.22330535884535179799266917387, 8.080804763659231577310421259525, 8.909066322925044918388895541115

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