↑

Properties

Label 2-430-43.6-c1-0-2
Degree $2$
Conductor $430$
Sign $-0.365 - 0.930i$
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  − 2-s + (−1.28 + 2.23i)3-s + 4-s + (0.5 − 0.866i)5-s + (1.28 − 2.23i)6-s + (2.24 + 3.88i)7-s − 8-s + (−1.82 − 3.16i)9-s + (−0.5 + 0.866i)10-s + 5.50·11-s + (−1.28 + 2.23i)12-s + (−1.37 − 2.37i)13-s + (−2.24 − 3.88i)14-s + (1.28 + 2.23i)15-s + 16-s + (3.07 + 5.31i)17-s + ⋯
L(s)  = 1  − 0.707·2-s + (−0.744 + 1.28i)3-s + 0.5·4-s + (0.223 − 0.387i)5-s + (0.526 − 0.911i)6-s + (0.847 + 1.46i)7-s − 0.353·8-s + (−0.608 − 1.05i)9-s + (−0.158 + 0.273i)10-s + 1.66·11-s + (−0.372 + 0.644i)12-s + (−0.380 − 0.659i)13-s + (−0.599 − 1.03i)14-s + (0.333 + 0.576i)15-s + 0.250·16-s + (0.744 + 1.28i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.365 - 0.930i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.365 - 0.930i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(430\)    =    \(2 \cdot 5 \cdot 43\)
Sign: $-0.365 - 0.930i$
Analytic conductor: \(3.43356\)
Root analytic conductor: \(1.85298\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{430} (221, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 430,\ (\ :1/2),\ -0.365 - 0.930i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.517229 + 0.759143i\)
\(L(\frac12)\) \(\approx\) \(0.517229 + 0.759143i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
5 \( 1 + (-0.5 + 0.866i)T \)
43 \( 1 + (5.43 - 3.66i)T \)
good3 \( 1 + (1.28 - 2.23i)T + (-1.5 - 2.59i)T^{2} \)
7 \( 1 + (-2.24 - 3.88i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 - 5.50T + 11T^{2} \)
13 \( 1 + (1.37 + 2.37i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-3.07 - 5.31i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.174 + 0.302i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.27 + 2.21i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.01 + 1.75i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (2.83 - 4.90i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.82 - 6.62i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 6.15T + 41T^{2} \)
47 \( 1 + 2.28T + 47T^{2} \)
53 \( 1 + (-7.08 + 12.2i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 6.14T + 59T^{2} \)
61 \( 1 + (-0.517 - 0.896i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.57 + 9.66i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-6.67 - 11.5i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-2.54 - 4.41i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (8.68 + 15.0i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (1.35 - 2.34i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-5.63 + 9.76i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 2.87T + 97T^{2} \)
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

−11.47963661812483031270267253932, −10.34249678994486067022209435092, −9.725228889949937439981370353387, −8.783416426979618002174836905795, −8.305621528735470815058864674220, −6.54109643435594111736105240791, −5.59233862882077988847997203429, −4.91226352562771824869607705478, −3.54673461417776904553684250256, −1.67860611201676728942352367848, 0.906347132305602258430712590551, 1.82115581825534098273844392265, 3.85721502398483425859832706280, 5.38661802893620301718635984200, 6.73016756533663545835486600028, 7.08710983602450572958216446354, 7.72024514779362992712404793793, 9.100120020231142499783855510803, 10.03526917339761043407732060944, 11.16583536161178741985615993094

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