Properties

Label 2-4600-5.4-c1-0-2
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  + 1.93i·3-s + 4.93i·7-s − 0.745·9-s + 0.745·11-s − 1.74i·13-s − 6.10i·17-s − 5.44·19-s − 9.55·21-s + i·23-s + 4.36i·27-s + 1.66·29-s − 1.61·31-s + 1.44i·33-s + 4.34i·37-s + 3.37·39-s + ⋯
L(s)  = 1  + 1.11i·3-s + 1.86i·7-s − 0.248·9-s + 0.224·11-s − 0.484i·13-s − 1.48i·17-s − 1.24·19-s − 2.08·21-s + 0.208i·23-s + 0.839i·27-s + 0.309·29-s − 0.290·31-s + 0.251i·33-s + 0.714i·37-s + 0.541·39-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

\(L(1)\) \(\approx\) \(0.4760178689\)
\(L(\frac12)\) \(\approx\) \(0.4760178689\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.93iT - 3T^{2} \)
7 \( 1 - 4.93iT - 7T^{2} \)
11 \( 1 - 0.745T + 11T^{2} \)
13 \( 1 + 1.74iT - 13T^{2} \)
17 \( 1 + 6.10iT - 17T^{2} \)
19 \( 1 + 5.44T + 19T^{2} \)
29 \( 1 - 1.66T + 29T^{2} \)
31 \( 1 + 1.61T + 31T^{2} \)
37 \( 1 - 4.34iT - 37T^{2} \)
41 \( 1 + 6.95T + 41T^{2} \)
43 \( 1 + 5.01iT - 43T^{2} \)
47 \( 1 - 2.68iT - 47T^{2} \)
53 \( 1 + 13.7iT - 53T^{2} \)
59 \( 1 + 12.2T + 59T^{2} \)
61 \( 1 + 13.9T + 61T^{2} \)
67 \( 1 - 13.1iT - 67T^{2} \)
71 \( 1 + 9.67T + 71T^{2} \)
73 \( 1 + 5.69iT - 73T^{2} \)
79 \( 1 + 10.3T + 79T^{2} \)
83 \( 1 + 0.637iT - 83T^{2} \)
89 \( 1 + 2.72T + 89T^{2} \)
97 \( 1 - 7.12iT - 97T^{2} \)
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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

−8.852725181981246116838660163508, −8.447158371472649501837890978148, −7.39211620806571498592970133249, −6.45469418068813699109999749919, −5.73445496213636607578610493764, −5.01153668558753905536227323973, −4.57021592440584243123759162702, −3.38626602287232560670770665231, −2.78180705870331648111531087966, −1.79603649955825930895827371337, 0.12489895260359317890238115803, 1.35162024356850711900845100528, 1.83047113678548038974272531737, 3.22685803951788473370190911113, 4.24226064313106523390457887425, 4.47095096408312255648799581633, 6.11274483258941312505808345948, 6.40705381377134358207069369889, 7.19271016416150135325467800361, 7.64510109453639092444884024778

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