Properties

Label 2-4600-5.4-c1-0-45
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  + 0.568i·3-s + 4.73i·7-s + 2.67·9-s − 0.360·11-s − 5.26i·13-s + 0.370i·17-s + 4.60·19-s − 2.69·21-s + i·23-s + 3.22i·27-s − 0.939·29-s + 9.66·31-s − 0.204i·33-s + 3.26i·37-s + 2.99·39-s + ⋯
L(s)  = 1  + 0.328i·3-s + 1.79i·7-s + 0.892·9-s − 0.108·11-s − 1.45i·13-s + 0.0899i·17-s + 1.05·19-s − 0.587·21-s + 0.208i·23-s + 0.620i·27-s − 0.174·29-s + 1.73·31-s − 0.0356i·33-s + 0.537i·37-s + 0.478·39-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

\(L(1)\) \(\approx\) \(2.238981472\)
\(L(\frac12)\) \(\approx\) \(2.238981472\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.568iT - 3T^{2} \)
7 \( 1 - 4.73iT - 7T^{2} \)
11 \( 1 + 0.360T + 11T^{2} \)
13 \( 1 + 5.26iT - 13T^{2} \)
17 \( 1 - 0.370iT - 17T^{2} \)
19 \( 1 - 4.60T + 19T^{2} \)
29 \( 1 + 0.939T + 29T^{2} \)
31 \( 1 - 9.66T + 31T^{2} \)
37 \( 1 - 3.26iT - 37T^{2} \)
41 \( 1 - 5.29T + 41T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 + 1.25iT - 47T^{2} \)
53 \( 1 + 10.9iT - 53T^{2} \)
59 \( 1 - 9.66T + 59T^{2} \)
61 \( 1 - 9.71T + 61T^{2} \)
67 \( 1 + 7.07iT - 67T^{2} \)
71 \( 1 - 11.3T + 71T^{2} \)
73 \( 1 + 0.745iT - 73T^{2} \)
79 \( 1 + 0.415T + 79T^{2} \)
83 \( 1 - 9.26iT - 83T^{2} \)
89 \( 1 + 12.6T + 89T^{2} \)
97 \( 1 - 14.0iT - 97T^{2} \)
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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

−8.253743110753255172469674860553, −8.044315269336039340087519542673, −6.94785576783405110504120938114, −6.16358570000952341017858954738, −5.25662538823497447292035797187, −5.12277116430642916696906814342, −3.83537983647318214582389277059, −2.97852489812261910437268698687, −2.28682162308717700497834461142, −1.01005192649590272653572104485, 0.799596821896219276811284577075, 1.48704176312984594711464798009, 2.68785470002435196219351418629, 4.01784645980985366313044812337, 4.16169505783866959518089330363, 5.07737246510119173959532469934, 6.30320815081325921067738701800, 6.90963032073149346713886975614, 7.35025056488297814509180894431, 7.933570529072419455437709867665

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