↑

Properties

Label 2-6300-21.20-c1-0-36
Degree $2$
Conductor $6300$
Sign $-0.746 + 0.665i$
Analytic cond. $50.3057$
Root an. cond. $7.09265$
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  + (−2.57 − 0.595i)7-s + 3.74i·11-s + 3.36i·13-s − 0.841·17-s + 5.59i·19-s + 2.35i·23-s + 1.41i·29-s − 8.66i·31-s − 5.15·37-s − 5.74·41-s − 3.32·43-s − 6.43·47-s + (6.29 + 3.06i)49-s − 9.64i·53-s + 12.2·59-s + ⋯
L(s)  = 1  + (−0.974 − 0.224i)7-s + 1.12i·11-s + 0.931i·13-s − 0.204·17-s + 1.28i·19-s + 0.490i·23-s + 0.262i·29-s − 1.55i·31-s − 0.847·37-s − 0.896·41-s − 0.507·43-s − 0.938·47-s + (0.898 + 0.438i)49-s − 1.32i·53-s + 1.59·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.746 + 0.665i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.746 + 0.665i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6300\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 7\)
Sign: $-0.746 + 0.665i$
Analytic conductor: \(50.3057\)
Root analytic conductor: \(7.09265\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{6300} (3401, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 6300,\ (\ :1/2),\ -0.746 + 0.665i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.02659455344\)
\(L(\frac12)\) \(\approx\) \(0.02659455344\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
7 \( 1 + (2.57 + 0.595i)T \)
good11 \( 1 - 3.74iT - 11T^{2} \)
13 \( 1 - 3.36iT - 13T^{2} \)
17 \( 1 + 0.841T + 17T^{2} \)
19 \( 1 - 5.59iT - 19T^{2} \)
23 \( 1 - 2.35iT - 23T^{2} \)
29 \( 1 - 1.41iT - 29T^{2} \)
31 \( 1 + 8.66iT - 31T^{2} \)
37 \( 1 + 5.15T + 37T^{2} \)
41 \( 1 + 5.74T + 41T^{2} \)
43 \( 1 + 3.32T + 43T^{2} \)
47 \( 1 + 6.43T + 47T^{2} \)
53 \( 1 + 9.64iT - 53T^{2} \)
59 \( 1 - 12.2T + 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 + 1.82T + 67T^{2} \)
71 \( 1 - 3.74iT - 71T^{2} \)
73 \( 1 + 0.979iT - 73T^{2} \)
79 \( 1 - 6.58T + 79T^{2} \)
83 \( 1 - 12.5T + 83T^{2} \)
89 \( 1 + 2.16T + 89T^{2} \)
97 \( 1 + 12.0iT - 97T^{2} \)
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

−7.69888324669923351387239518492, −6.91572041566263350206356165792, −6.56007454798613575601882961321, −5.67607115211973927393740181128, −4.86249058240116615269377692739, −3.96210793623985962396079047928, −3.50982646032961560828382853544, −2.29343825608532892787419963104, −1.58742496776852316296363175614, −0.00748647654251971262188644653, 0.971783077206042484850188364450, 2.43513787370551614641819154187, 3.14828507240745023596217718605, 3.66701062341593601443409592703, 4.91264708330038531318396100600, 5.40966061060288292146122141720, 6.36526164881618747502629707858, 6.69773576329331565815061100548, 7.58689216739819960014987265679, 8.548505426308465997797144943722

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