Properties

Label 2-4600-5.4-c1-0-15
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 1.83i·3-s + 3.97i·7-s − 0.384·9-s + 2.10·11-s + 5.35i·13-s − 1.29i·17-s − 2.10·19-s − 7.30·21-s i·23-s + 4.81i·27-s − 6.03·29-s − 8.32·31-s + 3.86i·33-s − 5.10i·37-s − 9.85·39-s + ⋯
L(s)  = 1  + 1.06i·3-s + 1.50i·7-s − 0.128·9-s + 0.634·11-s + 1.48i·13-s − 0.314i·17-s − 0.482·19-s − 1.59·21-s − 0.208i·23-s + 0.926i·27-s − 1.11·29-s − 1.49·31-s + 0.673i·33-s − 0.838i·37-s − 1.57·39-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

\(L(1)\) \(\approx\) \(1.320722590\)
\(L(\frac12)\) \(\approx\) \(1.320722590\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.83iT - 3T^{2} \)
7 \( 1 - 3.97iT - 7T^{2} \)
11 \( 1 - 2.10T + 11T^{2} \)
13 \( 1 - 5.35iT - 13T^{2} \)
17 \( 1 + 1.29iT - 17T^{2} \)
19 \( 1 + 2.10T + 19T^{2} \)
29 \( 1 + 6.03T + 29T^{2} \)
31 \( 1 + 8.32T + 31T^{2} \)
37 \( 1 + 5.10iT - 37T^{2} \)
41 \( 1 + 8.33T + 41T^{2} \)
43 \( 1 - 7.78iT - 43T^{2} \)
47 \( 1 + 11.3iT - 47T^{2} \)
53 \( 1 + 0.573iT - 53T^{2} \)
59 \( 1 - 9.17T + 59T^{2} \)
61 \( 1 - 13.5T + 61T^{2} \)
67 \( 1 - 15.8iT - 67T^{2} \)
71 \( 1 - 14.9T + 71T^{2} \)
73 \( 1 - 8.36iT - 73T^{2} \)
79 \( 1 - 9.41T + 79T^{2} \)
83 \( 1 - 1.51iT - 83T^{2} \)
89 \( 1 + 10.5T + 89T^{2} \)
97 \( 1 - 0.337iT - 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.850824284931532181543344861267, −8.392777450473058697972825351144, −7.05875340927609244250215467719, −6.63493736029612104557260005638, −5.47956076951387378961178767324, −5.21435530156251372997691965000, −4.04976630355794678360196001531, −3.75330219561334956552961528983, −2.42980949879126577265849039031, −1.74025146257455691680137716623, 0.36652519075668368638498649860, 1.23794814030020725084723139214, 2.08001246605113863262942439326, 3.45724220580118112726268976620, 3.89752492065479780799827205406, 4.99634076475822456263783545148, 5.85935980152266814740372095800, 6.73913473956267232942444178746, 7.11381192001301216508203816050, 7.85847184283244990664206651480

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