Properties

Label 2-3450-5.4-c1-0-30
Degree $2$
Conductor $3450$
Sign $0.894 + 0.447i$
Analytic cond. $27.5483$
Root an. cond. $5.24865$
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  i·2-s + i·3-s − 4-s + 6-s − 2i·7-s + i·8-s − 9-s + 2·11-s i·12-s + 2i·13-s − 2·14-s + 16-s + i·18-s − 8·19-s + 2·21-s − 2i·22-s + ⋯
L(s)  = 1  − 0.707i·2-s + 0.577i·3-s − 0.5·4-s + 0.408·6-s − 0.755i·7-s + 0.353i·8-s − 0.333·9-s + 0.603·11-s − 0.288i·12-s + 0.554i·13-s − 0.534·14-s + 0.250·16-s + 0.235i·18-s − 1.83·19-s + 0.436·21-s − 0.426i·22-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(3450\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 23\)
Sign: $0.894 + 0.447i$
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: \(0\)
Selberg data: \((2,\ 3450,\ (\ :1/2),\ 0.894 + 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.650111563\)
\(L(\frac12)\) \(\approx\) \(1.650111563\)
\(L(\frac{3}{2})\) 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 \( 1 + 2iT - 7T^{2} \)
11 \( 1 - 2T + 11T^{2} \)
13 \( 1 - 2iT - 13T^{2} \)
17 \( 1 - 17T^{2} \)
19 \( 1 + 8T + 19T^{2} \)
29 \( 1 - 10T + 29T^{2} \)
31 \( 1 - 8T + 31T^{2} \)
37 \( 1 - 8iT - 37T^{2} \)
41 \( 1 + 6T + 41T^{2} \)
43 \( 1 + 12iT - 43T^{2} \)
47 \( 1 - 8iT - 47T^{2} \)
53 \( 1 + 10iT - 53T^{2} \)
59 \( 1 + 4T + 59T^{2} \)
61 \( 1 - 12T + 61T^{2} \)
67 \( 1 + 4iT - 67T^{2} \)
71 \( 1 - 16T + 71T^{2} \)
73 \( 1 - 10iT - 73T^{2} \)
79 \( 1 + 10T + 79T^{2} \)
83 \( 1 - 10iT - 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 - 10iT - 97T^{2} \)
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

−8.453679605485522263187435528469, −8.281028368073281694480300016272, −6.74817882688283841819901377021, −6.53160607052570787030599976959, −5.21848510036479502162006803669, −4.37144490027030198185244195611, −4.02687175180789504037043217108, −3.00778413161477168460644940239, −1.99294127676939544676961041320, −0.77961067081901145361420462777, 0.74854409572395087390692142387, 2.11006488667088853878591070086, 2.99801721752752521504811295188, 4.21041166663859231026408996133, 4.92166332292426082433719553744, 5.97046187158078084221185552061, 6.35620876400494674680637922228, 7.01693371565295696275902382588, 8.049624222480126240166021194579, 8.491725685986546108368650146300

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