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Properties

Label 2-4600-5.4-c1-0-68
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3i·3-s + 2i·7-s − 6·9-s − 5i·13-s + 6i·17-s − 6·19-s − 6·21-s + i·23-s − 9i·27-s − 9·29-s + 3·31-s + 8i·37-s + 15·39-s + 3·41-s − 8i·43-s + ⋯
L(s)  = 1  + 1.73i·3-s + 0.755i·7-s − 2·9-s − 1.38i·13-s + 1.45i·17-s − 1.37·19-s − 1.30·21-s + 0.208i·23-s − 1.73i·27-s − 1.67·29-s + 0.538·31-s + 1.31i·37-s + 2.40·39-s + 0.468·41-s − 1.21i·43-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: \(1\)
Selberg data: \((2,\ 4600,\ (\ :1/2),\ 0.447 + 0.894i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 - 3iT - 3T^{2} \)
7 \( 1 - 2iT - 7T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 + 5iT - 13T^{2} \)
17 \( 1 - 6iT - 17T^{2} \)
19 \( 1 + 6T + 19T^{2} \)
29 \( 1 + 9T + 29T^{2} \)
31 \( 1 - 3T + 31T^{2} \)
37 \( 1 - 8iT - 37T^{2} \)
41 \( 1 - 3T + 41T^{2} \)
43 \( 1 + 8iT - 43T^{2} \)
47 \( 1 + 7iT - 47T^{2} \)
53 \( 1 + 2iT - 53T^{2} \)
59 \( 1 + 4T + 59T^{2} \)
61 \( 1 + 10T + 61T^{2} \)
67 \( 1 + 8iT - 67T^{2} \)
71 \( 1 - 7T + 71T^{2} \)
73 \( 1 - 9iT - 73T^{2} \)
79 \( 1 - 6T + 79T^{2} \)
83 \( 1 + 14iT - 83T^{2} \)
89 \( 1 + 16T + 89T^{2} \)
97 \( 1 + 6iT - 97T^{2} \)
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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

−8.468214902052962816709798002654, −7.77995806706178982265152148102, −6.43266421744379555071323253550, −5.71967753800840004977294169476, −5.29841655185222137949397838535, −4.36506737364489301577816132875, −3.71767671863564346959783769446, −3.00876975930645899658365000674, −1.97881782966950329025002164922, 0, 1.08831605253151223855013351164, 2.01664834644638459718230755768, 2.68444969132555361863695726691, 3.94898349350870437470422729977, 4.70519233851939142162232773833, 5.86510255429341595957904266917, 6.45117348550277483917142938497, 7.13302708999685900505024393488, 7.46878276599675555428658529202

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