Properties

Label 2-4600-5.4-c1-0-46
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  − 1.23i·3-s − 0.236i·7-s + 1.47·9-s + 11-s + 2.23i·13-s + 2.47i·17-s − 19-s − 0.291·21-s i·23-s − 5.52i·27-s + 6.23·29-s − 8.47·31-s − 1.23i·33-s − 6.76i·37-s + 2.76·39-s + ⋯
L(s)  = 1  − 0.713i·3-s − 0.0892i·7-s + 0.490·9-s + 0.301·11-s + 0.620i·13-s + 0.599i·17-s − 0.229·19-s − 0.0636·21-s − 0.208i·23-s − 1.06i·27-s + 1.15·29-s − 1.52·31-s − 0.215i·33-s − 1.11i·37-s + 0.442·39-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

\(L(1)\) \(\approx\) \(2.099200085\)
\(L(\frac12)\) \(\approx\) \(2.099200085\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.23iT - 3T^{2} \)
7 \( 1 + 0.236iT - 7T^{2} \)
11 \( 1 - T + 11T^{2} \)
13 \( 1 - 2.23iT - 13T^{2} \)
17 \( 1 - 2.47iT - 17T^{2} \)
19 \( 1 + T + 19T^{2} \)
29 \( 1 - 6.23T + 29T^{2} \)
31 \( 1 + 8.47T + 31T^{2} \)
37 \( 1 + 6.76iT - 37T^{2} \)
41 \( 1 - 11.9T + 41T^{2} \)
43 \( 1 - 11.4iT - 43T^{2} \)
47 \( 1 - 1.70iT - 47T^{2} \)
53 \( 1 + 3.23iT - 53T^{2} \)
59 \( 1 - 1.23T + 59T^{2} \)
61 \( 1 + 2.76T + 61T^{2} \)
67 \( 1 - 4.94iT - 67T^{2} \)
71 \( 1 - 10T + 71T^{2} \)
73 \( 1 + 0.527iT - 73T^{2} \)
79 \( 1 - 7.18T + 79T^{2} \)
83 \( 1 - 9iT - 83T^{2} \)
89 \( 1 + 2T + 89T^{2} \)
97 \( 1 - 16.1iT - 97T^{2} \)
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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

−8.063375004310325211272214583965, −7.56964521892321194976502745978, −6.75647617338326093496122084601, −6.32019320138899204081364672720, −5.43618571507306981409759320994, −4.34887953236486668901650709436, −3.89867415332916577072895792677, −2.61261581539510535648244937039, −1.79175041630982347049259360433, −0.861547556275380727622899041899, 0.799973499718488600341583529512, 2.05923097985491123177103830134, 3.12389559337084299398196648694, 3.87829425719174037918239165735, 4.63355156205046568612357614609, 5.32588731330798962974952497215, 6.09650644538525204344091050331, 7.05364703761660505664492770827, 7.54720235534033804938995379990, 8.525822970655799763427615386917

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