↑

Properties

Label 2-5850-5.4-c1-0-44
Degree $2$
Conductor $5850$
Sign $0.894 - 0.447i$
Analytic cond. $46.7124$
Root an. cond. $6.83465$
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 − 4-s + 4i·7-s − i·8-s − 4·11-s + i·13-s − 4·14-s + 16-s − 4i·17-s − 7·19-s − 4i·22-s − 4i·23-s − 26-s − 4i·28-s + 5·29-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.5·4-s + 1.51i·7-s − 0.353i·8-s − 1.20·11-s + 0.277i·13-s − 1.06·14-s + 0.250·16-s − 0.970i·17-s − 1.60·19-s − 0.852i·22-s − 0.834i·23-s − 0.196·26-s − 0.755i·28-s + 0.928·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5850\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 13\)
Sign: $0.894 - 0.447i$
Analytic conductor: \(46.7124\)
Root analytic conductor: \(6.83465\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{5850} (5149, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 5850,\ (\ :1/2),\ 0.894 - 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.186165935\)
\(L(\frac12)\) \(\approx\) \(1.186165935\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 \)
13 \( 1 - iT \)
good7 \( 1 - 4iT - 7T^{2} \)
11 \( 1 + 4T + 11T^{2} \)
17 \( 1 + 4iT - 17T^{2} \)
19 \( 1 + 7T + 19T^{2} \)
23 \( 1 + 4iT - 23T^{2} \)
29 \( 1 - 5T + 29T^{2} \)
31 \( 1 - 4T + 31T^{2} \)
37 \( 1 + 9iT - 37T^{2} \)
41 \( 1 - 5T + 41T^{2} \)
43 \( 1 + 10iT - 43T^{2} \)
47 \( 1 - 3iT - 47T^{2} \)
53 \( 1 + 9iT - 53T^{2} \)
59 \( 1 + 6T + 59T^{2} \)
61 \( 1 - 4T + 61T^{2} \)
67 \( 1 - 7iT - 67T^{2} \)
71 \( 1 - 15T + 71T^{2} \)
73 \( 1 - 12iT - 73T^{2} \)
79 \( 1 + 7T + 79T^{2} \)
83 \( 1 + 6iT - 83T^{2} \)
89 \( 1 - 14T + 89T^{2} \)
97 \( 1 - 16iT - 97T^{2} \)
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

−8.243478933449005771695250647339, −7.48232618745503085101824845354, −6.60009352984185316534604411728, −6.09047244804108930750696249523, −5.25781921798940436621876961714, −4.85590294467844072164906292850, −3.87257032437046244434758191368, −2.49948778719731214368567975365, −2.36608322842577153953136247642, −0.42737535465808226092387759004, 0.74102378561572624346633119564, 1.73461133612842798534215969006, 2.78982880721431412424437779256, 3.53242218575775701644803547587, 4.44362316588805938081034867135, 4.76562177513051621704919198862, 5.97235922309932538654245930368, 6.57078748440138832713359759762, 7.58522245813205803223718578175, 8.047201660933357820308696279043

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