↑

Properties

Label 2-430-43.13-c1-0-1
Degree $2$
Conductor $430$
Sign $-0.966 + 0.256i$
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  + (−0.222 + 0.974i)2-s + (−1.04 + 0.968i)3-s + (−0.900 − 0.433i)4-s + (0.988 + 0.149i)5-s + (−0.711 − 1.23i)6-s + (0.109 − 0.188i)7-s + (0.623 − 0.781i)8-s + (−0.0727 + 0.970i)9-s + (−0.365 + 0.930i)10-s + (−5.00 + 2.41i)11-s + (1.36 − 0.419i)12-s + (0.0151 + 0.0387i)13-s + (0.159 + 0.148i)14-s + (−1.17 + 0.802i)15-s + (0.623 + 0.781i)16-s + (−2.71 + 0.409i)17-s + ⋯
L(s)  = 1  + (−0.157 + 0.689i)2-s + (−0.602 + 0.559i)3-s + (−0.450 − 0.216i)4-s + (0.442 + 0.0666i)5-s + (−0.290 − 0.503i)6-s + (0.0412 − 0.0714i)7-s + (0.220 − 0.276i)8-s + (−0.0242 + 0.323i)9-s + (−0.115 + 0.294i)10-s + (−1.50 + 0.727i)11-s + (0.392 − 0.121i)12-s + (0.00421 + 0.0107i)13-s + (0.0427 + 0.0396i)14-s + (−0.303 + 0.207i)15-s + (0.155 + 0.195i)16-s + (−0.659 + 0.0994i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.966 + 0.256i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.966 + 0.256i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(430\)    =    \(2 \cdot 5 \cdot 43\)
Sign: $-0.966 + 0.256i$
Analytic conductor: \(3.43356\)
Root analytic conductor: \(1.85298\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{430} (271, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 430,\ (\ :1/2),\ -0.966 + 0.256i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0722443 - 0.552988i\)
\(L(\frac12)\) \(\approx\) \(0.0722443 - 0.552988i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.222 - 0.974i)T \)
5 \( 1 + (-0.988 - 0.149i)T \)
43 \( 1 + (-6.38 - 1.51i)T \)
good3 \( 1 + (1.04 - 0.968i)T + (0.224 - 2.99i)T^{2} \)
7 \( 1 + (-0.109 + 0.188i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (5.00 - 2.41i)T + (6.85 - 8.60i)T^{2} \)
13 \( 1 + (-0.0151 - 0.0387i)T + (-9.52 + 8.84i)T^{2} \)
17 \( 1 + (2.71 - 0.409i)T + (16.2 - 5.01i)T^{2} \)
19 \( 1 + (-0.212 - 2.83i)T + (-18.7 + 2.83i)T^{2} \)
23 \( 1 + (0.990 + 0.675i)T + (8.40 + 21.4i)T^{2} \)
29 \( 1 + (0.449 + 0.416i)T + (2.16 + 28.9i)T^{2} \)
31 \( 1 + (7.03 - 2.17i)T + (25.6 - 17.4i)T^{2} \)
37 \( 1 + (2.45 + 4.24i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-0.188 + 0.827i)T + (-36.9 - 17.7i)T^{2} \)
47 \( 1 + (10.3 + 5.00i)T + (29.3 + 36.7i)T^{2} \)
53 \( 1 + (-0.231 + 0.588i)T + (-38.8 - 36.0i)T^{2} \)
59 \( 1 + (-9.15 - 11.4i)T + (-13.1 + 57.5i)T^{2} \)
61 \( 1 + (-11.5 - 3.56i)T + (50.4 + 34.3i)T^{2} \)
67 \( 1 + (-0.474 - 6.32i)T + (-66.2 + 9.98i)T^{2} \)
71 \( 1 + (10.1 - 6.92i)T + (25.9 - 66.0i)T^{2} \)
73 \( 1 + (-2.92 - 7.45i)T + (-53.5 + 49.6i)T^{2} \)
79 \( 1 + (0.192 - 0.333i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-6.71 + 6.23i)T + (6.20 - 82.7i)T^{2} \)
89 \( 1 + (0.469 - 0.435i)T + (6.65 - 88.7i)T^{2} \)
97 \( 1 + (-1.87 + 0.903i)T + (60.4 - 75.8i)T^{2} \)
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

−11.39355380865742245648984430235, −10.40781969862480917764313086536, −10.11426997052992513051936848389, −8.922655099366241524656773225452, −7.86924569297050285826215854686, −7.05185437947255472234609068404, −5.73446063291728820221155946639, −5.21329293215804340596739937344, −4.16208378712414151396335551791, −2.25304995537753034730190105797, 0.38004218021648101772912526189, 2.08610029809131597473439058233, 3.34973858234779268851496974729, 4.97628010099779184436101347688, 5.79696142432293820872996530545, 6.89186186873299688947982793267, 8.006104943467719457863359938098, 8.977675313227117575485411087754, 9.890592876980803077134569491542, 10.94885498137771645576733705419

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