Properties

Label 2-3450-5.4-c1-0-51
Degree $2$
Conductor $3450$
Sign $0.447 + 0.894i$
Analytic cond. $27.5483$
Root an. cond. $5.24865$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + i·2-s + i·3-s − 4-s − 6-s + 3i·7-s i·8-s − 9-s − 3·11-s i·12-s + 3i·13-s − 3·14-s + 16-s − 4i·17-s i·18-s + 3·19-s + ⋯
L(s)  = 1  + 0.707i·2-s + 0.577i·3-s − 0.5·4-s − 0.408·6-s + 1.13i·7-s − 0.353i·8-s − 0.333·9-s − 0.904·11-s − 0.288i·12-s + 0.832i·13-s − 0.801·14-s + 0.250·16-s − 0.970i·17-s − 0.235i·18-s + 0.688·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3450\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 23\)
Sign: $0.447 + 0.894i$
Analytic conductor: \(27.5483\)
Root analytic conductor: \(5.24865\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3450} (2899, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 3450,\ (\ :1/2),\ 0.447 + 0.894i)\)

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 \( 1 - iT \)
3 \( 1 - iT \)
5 \( 1 \)
23 \( 1 - iT \)
good7 \( 1 - 3iT - 7T^{2} \)
11 \( 1 + 3T + 11T^{2} \)
13 \( 1 - 3iT - 13T^{2} \)
17 \( 1 + 4iT - 17T^{2} \)
19 \( 1 - 3T + 19T^{2} \)
29 \( 1 + 3T + 29T^{2} \)
31 \( 1 + 10T + 31T^{2} \)
37 \( 1 + 8iT - 37T^{2} \)
41 \( 1 + 9T + 41T^{2} \)
43 \( 1 + iT - 43T^{2} \)
47 \( 1 - 2iT - 47T^{2} \)
53 \( 1 - 6iT - 53T^{2} \)
59 \( 1 + 8T + 59T^{2} \)
61 \( 1 - 12T + 61T^{2} \)
67 \( 1 - 8iT - 67T^{2} \)
71 \( 1 + 14T + 71T^{2} \)
73 \( 1 + 13iT - 73T^{2} \)
79 \( 1 - 17T + 79T^{2} \)
83 \( 1 + iT - 83T^{2} \)
89 \( 1 - 6T + 89T^{2} \)
97 \( 1 - 6iT - 97T^{2} \)
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

−8.594727561096250568607914683950, −7.60604083458021867815907882228, −7.12240693561778877532409930352, −6.00909156403341296823250949620, −5.39423309486182384637904724093, −4.95355972115712410778498567211, −3.87948129659742253858868981963, −2.95670951281968131037554877040, −1.96010050643571310223846467719, 0, 1.13531973161842769993227485333, 2.09687067171718710887999088900, 3.24398825784547830860069987967, 3.77193710054615342007457735688, 4.95156745147568085045581984651, 5.54860256187141924715838265477, 6.58193992554733461097521977922, 7.42642995809189757533603819840, 7.947742416903719463836030478211

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