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Properties

Label 2-430-43.24-c1-0-7
Degree $2$
Conductor $430$
Sign $-0.518 + 0.855i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.623 − 0.781i)2-s + (−2.81 + 0.424i)3-s + (−0.222 − 0.974i)4-s + (0.826 − 0.563i)5-s + (−1.42 + 2.46i)6-s + (1.01 + 1.75i)7-s + (−0.900 − 0.433i)8-s + (4.90 − 1.51i)9-s + (0.0747 − 0.997i)10-s + (0.219 − 0.962i)11-s + (1.04 + 2.65i)12-s + (−0.407 − 5.44i)13-s + (2.00 + 0.301i)14-s + (−2.09 + 1.93i)15-s + (−0.900 + 0.433i)16-s + (−5.20 − 3.55i)17-s + ⋯
L(s)  = 1  + (0.440 − 0.552i)2-s + (−1.62 + 0.245i)3-s + (−0.111 − 0.487i)4-s + (0.369 − 0.251i)5-s + (−0.581 + 1.00i)6-s + (0.382 + 0.663i)7-s + (−0.318 − 0.153i)8-s + (1.63 − 0.503i)9-s + (0.0236 − 0.315i)10-s + (0.0662 − 0.290i)11-s + (0.300 + 0.766i)12-s + (−0.113 − 1.50i)13-s + (0.535 + 0.0806i)14-s + (−0.539 + 0.500i)15-s + (−0.225 + 0.108i)16-s + (−1.26 − 0.861i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.518 + 0.855i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.518 + 0.855i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(430\)    =    \(2 \cdot 5 \cdot 43\)
Sign: $-0.518 + 0.855i$
Analytic conductor: \(3.43356\)
Root analytic conductor: \(1.85298\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{430} (411, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 430,\ (\ :1/2),\ -0.518 + 0.855i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.424552 - 0.753916i\)
\(L(\frac12)\) \(\approx\) \(0.424552 - 0.753916i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.623 + 0.781i)T \)
5 \( 1 + (-0.826 + 0.563i)T \)
43 \( 1 + (4.16 - 5.06i)T \)
good3 \( 1 + (2.81 - 0.424i)T + (2.86 - 0.884i)T^{2} \)
7 \( 1 + (-1.01 - 1.75i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.219 + 0.962i)T + (-9.91 - 4.77i)T^{2} \)
13 \( 1 + (0.407 + 5.44i)T + (-12.8 + 1.93i)T^{2} \)
17 \( 1 + (5.20 + 3.55i)T + (6.21 + 15.8i)T^{2} \)
19 \( 1 + (-2.03 - 0.628i)T + (15.6 + 10.7i)T^{2} \)
23 \( 1 + (3.57 + 3.31i)T + (1.71 + 22.9i)T^{2} \)
29 \( 1 + (-1.59 - 0.240i)T + (27.7 + 8.54i)T^{2} \)
31 \( 1 + (2.85 + 7.27i)T + (-22.7 + 21.0i)T^{2} \)
37 \( 1 + (-4.45 + 7.71i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (1.05 - 1.32i)T + (-9.12 - 39.9i)T^{2} \)
47 \( 1 + (-1.28 - 5.62i)T + (-42.3 + 20.3i)T^{2} \)
53 \( 1 + (0.0781 - 1.04i)T + (-52.4 - 7.89i)T^{2} \)
59 \( 1 + (-13.1 + 6.31i)T + (36.7 - 46.1i)T^{2} \)
61 \( 1 + (-2.97 + 7.57i)T + (-44.7 - 41.4i)T^{2} \)
67 \( 1 + (-6.37 - 1.96i)T + (55.3 + 37.7i)T^{2} \)
71 \( 1 + (2.83 - 2.63i)T + (5.30 - 70.8i)T^{2} \)
73 \( 1 + (-0.300 - 4.00i)T + (-72.1 + 10.8i)T^{2} \)
79 \( 1 + (-7.31 - 12.6i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (11.7 - 1.77i)T + (79.3 - 24.4i)T^{2} \)
89 \( 1 + (7.77 - 1.17i)T + (85.0 - 26.2i)T^{2} \)
97 \( 1 + (0.963 - 4.22i)T + (-87.3 - 42.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.21971811948845402906185060611, −10.20945734386542382222001830327, −9.461511503129446280509353177051, −8.158430801022974317606907539607, −6.66031391797315305337143900936, −5.67019850508765671275093826858, −5.27242524454915173342954114922, −4.22109573459103167360644280818, −2.45047316159219936071893435855, −0.59225842148009159028718707398, 1.71793613313696505841790396272, 4.07076067062900722688742217112, 4.84817166834179403023961664338, 5.86789258634855094153199699481, 6.79254943508749367145846773744, 7.12399822684234034647568490469, 8.620201515533372041601829227732, 9.913721297041999066332974207841, 10.79272679418429322696358726860, 11.58483318590706900467193519092

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