Properties

Label 2-3450-5.4-c1-0-18
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 + 5.12i·7-s + i·8-s − 9-s + 5.12·11-s + i·12-s + 2i·13-s + 5.12·14-s + 16-s − 7.12i·17-s + i·18-s − 4·19-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.577i·3-s − 0.5·4-s − 0.408·6-s + 1.93i·7-s + 0.353i·8-s − 0.333·9-s + 1.54·11-s + 0.288i·12-s + 0.554i·13-s + 1.36·14-s + 0.250·16-s − 1.72i·17-s + 0.235i·18-s − 0.917·19-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

\(L(1)\) \(\approx\) \(1.507645832\)
\(L(\frac12)\) \(\approx\) \(1.507645832\)
\(L(\frac{3}{2})\) 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 \( 1 - 5.12iT - 7T^{2} \)
11 \( 1 - 5.12T + 11T^{2} \)
13 \( 1 - 2iT - 13T^{2} \)
17 \( 1 + 7.12iT - 17T^{2} \)
19 \( 1 + 4T + 19T^{2} \)
29 \( 1 + 2T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 - 7.12iT - 37T^{2} \)
41 \( 1 - 2T + 41T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 - 8iT - 47T^{2} \)
53 \( 1 + 4.24iT - 53T^{2} \)
59 \( 1 - 14.2T + 59T^{2} \)
61 \( 1 - 0.876T + 61T^{2} \)
67 \( 1 - 8iT - 67T^{2} \)
71 \( 1 - 6.24T + 71T^{2} \)
73 \( 1 - 12.2iT - 73T^{2} \)
79 \( 1 - 5.12T + 79T^{2} \)
83 \( 1 - 11.3iT - 83T^{2} \)
89 \( 1 + 3.12T + 89T^{2} \)
97 \( 1 + 0.246iT - 97T^{2} \)
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

−8.722457995193821617594679643176, −8.251046633550159255519468716978, −6.99147557093146811223111561719, −6.43783773808062316460392684465, −5.60136926833259158165553973156, −4.86170573229581804429762824791, −3.84750622133712027756308148593, −2.74730712632429857836956326375, −2.20664352222468011985422042775, −1.19027299419765442153715000695, 0.50093876815806335363385456539, 1.68485164808227681989585066417, 3.58033405128142892823340861114, 3.95593281602730264158125172457, 4.46056600830069372358049403947, 5.63063195866398608836386577653, 6.45832428338448985155961531899, 6.91681530437523969318487135242, 7.81636017929584628300719789821, 8.426751138366929919528912567523

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