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Properties

Label 2-430-43.24-c1-0-12
Degree $2$
Conductor $430$
Sign $-0.875 + 0.483i$
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 + (−0.933 + 0.140i)3-s + (−0.222 − 0.974i)4-s + (−0.826 + 0.563i)5-s + (−0.472 + 0.817i)6-s + (−0.0334 − 0.0579i)7-s + (−0.900 − 0.433i)8-s + (−2.01 + 0.621i)9-s + (−0.0747 + 0.997i)10-s + (1.09 − 4.77i)11-s + (0.344 + 0.879i)12-s + (−0.403 − 5.38i)13-s + (−0.0661 − 0.00997i)14-s + (0.692 − 0.642i)15-s + (−0.900 + 0.433i)16-s + (−1.58 − 1.08i)17-s + ⋯
L(s)  = 1  + (0.440 − 0.552i)2-s + (−0.539 + 0.0812i)3-s + (−0.111 − 0.487i)4-s + (−0.369 + 0.251i)5-s + (−0.192 + 0.333i)6-s + (−0.0126 − 0.0219i)7-s + (−0.318 − 0.153i)8-s + (−0.671 + 0.207i)9-s + (−0.0236 + 0.315i)10-s + (0.328 − 1.44i)11-s + (0.0995 + 0.253i)12-s + (−0.112 − 1.49i)13-s + (−0.0176 − 0.00266i)14-s + (0.178 − 0.165i)15-s + (−0.225 + 0.108i)16-s + (−0.384 − 0.262i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.875 + 0.483i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.875 + 0.483i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(430\)    =    \(2 \cdot 5 \cdot 43\)
Sign: $-0.875 + 0.483i$
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: \((2,\ 430,\ (\ :1/2),\ -0.875 + 0.483i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.202165 - 0.783390i\)
\(L(\frac12)\) \(\approx\) \(0.202165 - 0.783390i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-6.13 + 2.31i)T \)
good3 \( 1 + (0.933 - 0.140i)T + (2.86 - 0.884i)T^{2} \)
7 \( 1 + (0.0334 + 0.0579i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.09 + 4.77i)T + (-9.91 - 4.77i)T^{2} \)
13 \( 1 + (0.403 + 5.38i)T + (-12.8 + 1.93i)T^{2} \)
17 \( 1 + (1.58 + 1.08i)T + (6.21 + 15.8i)T^{2} \)
19 \( 1 + (7.78 + 2.40i)T + (15.6 + 10.7i)T^{2} \)
23 \( 1 + (-2.10 - 1.95i)T + (1.71 + 22.9i)T^{2} \)
29 \( 1 + (4.55 + 0.687i)T + (27.7 + 8.54i)T^{2} \)
31 \( 1 + (-3.88 - 9.90i)T + (-22.7 + 21.0i)T^{2} \)
37 \( 1 + (-0.345 + 0.598i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.670 - 0.840i)T + (-9.12 - 39.9i)T^{2} \)
47 \( 1 + (2.51 + 11.0i)T + (-42.3 + 20.3i)T^{2} \)
53 \( 1 + (0.387 - 5.16i)T + (-52.4 - 7.89i)T^{2} \)
59 \( 1 + (4.37 - 2.10i)T + (36.7 - 46.1i)T^{2} \)
61 \( 1 + (-2.11 + 5.38i)T + (-44.7 - 41.4i)T^{2} \)
67 \( 1 + (-10.7 - 3.32i)T + (55.3 + 37.7i)T^{2} \)
71 \( 1 + (7.29 - 6.77i)T + (5.30 - 70.8i)T^{2} \)
73 \( 1 + (1.06 + 14.1i)T + (-72.1 + 10.8i)T^{2} \)
79 \( 1 + (-7.43 - 12.8i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-5.84 + 0.881i)T + (79.3 - 24.4i)T^{2} \)
89 \( 1 + (8.54 - 1.28i)T + (85.0 - 26.2i)T^{2} \)
97 \( 1 + (2.00 - 8.78i)T + (-87.3 - 42.0i)T^{2} \)
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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

−10.86371286781637157729082029651, −10.43761165759124957830225778134, −8.874615116161147741931453312714, −8.256794623759359422320766845843, −6.76630741742809015931698209393, −5.82535344888527268209612139892, −5.02264254099067280288197501730, −3.63759245364381247005887044806, −2.68626745008799409369301586584, −0.45720546679863712824983504321, 2.18228079530358649076525123590, 4.13931547461272657137466569350, 4.60950103616500912181033286436, 6.06997991741386794295005132955, 6.65154072298351482942622678644, 7.68931883345354211552745094363, 8.781313946982477520300497062894, 9.534500225580882706325576983303, 10.92551131256833517436710605208, 11.67557441048032580428041932252

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