Properties

Label 2-4600-5.4-c1-0-28
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  + 0.523i·3-s + 0.476i·7-s + 2.72·9-s − 1.67·11-s − 2.67i·13-s + 1.67i·17-s − 7.92·19-s − 0.249·21-s i·23-s + 3i·27-s + 2.20·29-s + 6.77·31-s − 0.878i·33-s + 4i·37-s + 1.40·39-s + ⋯
L(s)  = 1  + 0.302i·3-s + 0.179i·7-s + 0.908·9-s − 0.505·11-s − 0.742i·13-s + 0.406i·17-s − 1.81·19-s − 0.0544·21-s − 0.208i·23-s + 0.577i·27-s + 0.408·29-s + 1.21·31-s − 0.153i·33-s + 0.657i·37-s + 0.224·39-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

\(L(1)\) \(\approx\) \(1.737179874\)
\(L(\frac12)\) \(\approx\) \(1.737179874\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.523iT - 3T^{2} \)
7 \( 1 - 0.476iT - 7T^{2} \)
11 \( 1 + 1.67T + 11T^{2} \)
13 \( 1 + 2.67iT - 13T^{2} \)
17 \( 1 - 1.67iT - 17T^{2} \)
19 \( 1 + 7.92T + 19T^{2} \)
29 \( 1 - 2.20T + 29T^{2} \)
31 \( 1 - 6.77T + 31T^{2} \)
37 \( 1 - 4iT - 37T^{2} \)
41 \( 1 - 5.97T + 41T^{2} \)
43 \( 1 - 0.402iT - 43T^{2} \)
47 \( 1 - 1.79iT - 47T^{2} \)
53 \( 1 - 10.8iT - 53T^{2} \)
59 \( 1 + 9.45T + 59T^{2} \)
61 \( 1 - 6.32T + 61T^{2} \)
67 \( 1 - 14.7iT - 67T^{2} \)
71 \( 1 - 9.97T + 71T^{2} \)
73 \( 1 - 6.10iT - 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + 4.49iT - 83T^{2} \)
89 \( 1 - 3.59T + 89T^{2} \)
97 \( 1 + 6.08iT - 97T^{2} \)
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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

−8.366248885584651390353399409458, −7.86940173031690029214670580375, −6.95551469414767415003939051119, −6.26478752072980560455082236433, −5.53421138388121904486451895444, −4.51936009156142980338598578972, −4.17468367007585049028268617853, −2.99221611461249625185523631033, −2.20613586778917673374422232963, −0.971890300648105959864326566294, 0.56559579865520462447109211236, 1.83690412465453070576025163005, 2.51446714722516750308001293744, 3.78630080699195349151673022169, 4.42919763886811375532492091587, 5.11494389817308811311501058760, 6.32048935458583879976922409977, 6.62961224463539028750563335382, 7.49808677560816723647680434594, 8.082784173110893796924619425651

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