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Properties

Label 2-430-215.49-c1-0-19
Degree $2$
Conductor $430$
Sign $-0.696 + 0.717i$
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  + i·2-s + (−1.09 − 0.631i)3-s − 4-s + (1.69 − 1.45i)5-s + (0.631 − 1.09i)6-s + (−2.36 + 1.36i)7-s − i·8-s + (−0.701 − 1.21i)9-s + (1.45 + 1.69i)10-s − 3.35·11-s + (1.09 + 0.631i)12-s + (−0.651 + 0.376i)13-s + (−1.36 − 2.36i)14-s + (−2.77 + 0.518i)15-s + 16-s + (−2.73 + 1.57i)17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (−0.631 − 0.364i)3-s − 0.5·4-s + (0.759 − 0.650i)5-s + (0.257 − 0.446i)6-s + (−0.894 + 0.516i)7-s − 0.353i·8-s + (−0.233 − 0.405i)9-s + (0.459 + 0.537i)10-s − 1.01·11-s + (0.315 + 0.182i)12-s + (−0.180 + 0.104i)13-s + (−0.365 − 0.632i)14-s + (−0.717 + 0.133i)15-s + 0.250·16-s + (−0.663 + 0.382i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.696 + 0.717i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.696 + 0.717i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(430\)    =    \(2 \cdot 5 \cdot 43\)
Sign: $-0.696 + 0.717i$
Analytic conductor: \(3.43356\)
Root analytic conductor: \(1.85298\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{430} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 430,\ (\ :1/2),\ -0.696 + 0.717i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0927615 - 0.219157i\)
\(L(\frac12)\) \(\approx\) \(0.0927615 - 0.219157i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
5 \( 1 + (-1.69 + 1.45i)T \)
43 \( 1 + (-1.38 + 6.41i)T \)
good3 \( 1 + (1.09 + 0.631i)T + (1.5 + 2.59i)T^{2} \)
7 \( 1 + (2.36 - 1.36i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + 3.35T + 11T^{2} \)
13 \( 1 + (0.651 - 0.376i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (2.73 - 1.57i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.51 - 2.62i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (5.67 + 3.27i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.41 + 5.91i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.827 + 1.43i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (5.85 + 3.37i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 7.42T + 41T^{2} \)
47 \( 1 + 2.79iT - 47T^{2} \)
53 \( 1 + (-4.38 - 2.52i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + 6.16T + 59T^{2} \)
61 \( 1 + (-4.05 - 7.02i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.66 + 2.11i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-3.62 - 6.28i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (1.16 - 0.671i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-7.44 - 12.8i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-4.04 - 2.33i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (-0.295 + 0.511i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 11.6iT - 97T^{2} \)
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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

−10.66301335199930876772408100467, −9.771353035182999719023880943640, −8.971246871148059843330773114064, −8.092104612695293186236006134531, −6.79963915434992290584975655784, −5.91377664417177668629908431332, −5.59022275115523320758101159191, −4.13581281272751191095152875363, −2.32481292962678593148901998648, −0.14941853392979837785708922125, 2.25938795922757698175646592833, 3.31054272223931589158265410821, 4.75938673491288639932152002188, 5.67178255620734009422825194758, 6.67528911468587575552098058431, 7.80979876836346020266898284500, 9.226305796629715952351631682289, 9.985621869096747281026406722819, 10.67855670547739111509775761558, 11.10145512684260379337134133220

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