↑

Properties

Label 2-430-215.128-c1-0-15
Degree $2$
Conductor $430$
Sign $0.436 + 0.899i$
Analytic cond. $3.43356$
Root an. cond. $1.85298$
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  + (0.707 + 0.707i)2-s + (−2.10 + 2.10i)3-s + 1.00i·4-s + (1.02 − 1.98i)5-s − 2.97·6-s + (−3.26 − 3.26i)7-s + (−0.707 + 0.707i)8-s − 5.85i·9-s + (2.13 − 0.675i)10-s + 2.00·11-s + (−2.10 − 2.10i)12-s + (−3.38 − 3.38i)13-s − 4.61i·14-s + (2.00 + 6.34i)15-s − 1.00·16-s + (0.281 − 0.281i)17-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + (−1.21 + 1.21i)3-s + 0.500i·4-s + (0.460 − 0.887i)5-s − 1.21·6-s + (−1.23 − 1.23i)7-s + (−0.250 + 0.250i)8-s − 1.95i·9-s + (0.674 − 0.213i)10-s + 0.605·11-s + (−0.607 − 0.607i)12-s + (−0.938 − 0.938i)13-s − 1.23i·14-s + (0.518 + 1.63i)15-s − 0.250·16-s + (0.0683 − 0.0683i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.436 + 0.899i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.436 + 0.899i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(430\)    =    \(2 \cdot 5 \cdot 43\)
Sign: $0.436 + 0.899i$
Analytic conductor: \(3.43356\)
Root analytic conductor: \(1.85298\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{430} (343, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 430,\ (\ :1/2),\ 0.436 + 0.899i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.481167 - 0.301510i\)
\(L(\frac12)\) \(\approx\) \(0.481167 - 0.301510i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 - 0.707i)T \)
5 \( 1 + (-1.02 + 1.98i)T \)
43 \( 1 + (4.21 + 5.02i)T \)
good3 \( 1 + (2.10 - 2.10i)T - 3iT^{2} \)
7 \( 1 + (3.26 + 3.26i)T + 7iT^{2} \)
11 \( 1 - 2.00T + 11T^{2} \)
13 \( 1 + (3.38 + 3.38i)T + 13iT^{2} \)
17 \( 1 + (-0.281 + 0.281i)T - 17iT^{2} \)
19 \( 1 + 8.17T + 19T^{2} \)
23 \( 1 + (-2.02 - 2.02i)T + 23iT^{2} \)
29 \( 1 - 6.79T + 29T^{2} \)
31 \( 1 - 5.21T + 31T^{2} \)
37 \( 1 + (-0.298 - 0.298i)T + 37iT^{2} \)
41 \( 1 + 7.03T + 41T^{2} \)
47 \( 1 + (5.34 - 5.34i)T - 47iT^{2} \)
53 \( 1 + (6.37 + 6.37i)T + 53iT^{2} \)
59 \( 1 + 7.42iT - 59T^{2} \)
61 \( 1 + 10.7iT - 61T^{2} \)
67 \( 1 + (0.771 - 0.771i)T - 67iT^{2} \)
71 \( 1 + 3.48iT - 71T^{2} \)
73 \( 1 + (2.57 - 2.57i)T - 73iT^{2} \)
79 \( 1 - 6.91iT - 79T^{2} \)
83 \( 1 + (-0.858 - 0.858i)T + 83iT^{2} \)
89 \( 1 - 8.21T + 89T^{2} \)
97 \( 1 + (2.39 - 2.39i)T - 97iT^{2} \)
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

−10.81568932319686031844990818912, −10.04550277484091822380954632904, −9.592173968942716878885319783381, −8.308011206839893545337811230375, −6.69837901319550633021890745211, −6.25051149015660135677130140083, −5.03043440684709702505149660425, −4.46288414587294682080920788438, −3.43195200549905977965232305891, −0.33641253438206179322080419596, 1.94642513014574193015304036065, 2.83109256238150619640654771971, 4.71296925673734311628224082219, 6.02191279378604024112945805014, 6.44319937237648525791357356734, 6.93580941719048331730260960585, 8.701280380073500063670064681822, 9.852787793875086502335005949669, 10.61623493455600445402854592411, 11.77424197012106320877148063807

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