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Properties

Label 2-430-43.24-c1-0-11
Degree $2$
Conductor $430$
Sign $-0.215 + 0.976i$
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 + (0.118 − 0.0178i)3-s + (−0.222 − 0.974i)4-s + (0.826 − 0.563i)5-s + (−0.0599 + 0.103i)6-s + (−1.86 − 3.22i)7-s + (0.900 + 0.433i)8-s + (−2.85 + 0.880i)9-s + (−0.0747 + 0.997i)10-s + (0.258 − 1.13i)11-s + (−0.0437 − 0.111i)12-s + (0.0534 + 0.713i)13-s + (3.68 + 0.554i)14-s + (0.0878 − 0.0815i)15-s + (−0.900 + 0.433i)16-s + (−5.36 − 3.65i)17-s + ⋯
L(s)  = 1  + (−0.440 + 0.552i)2-s + (0.0684 − 0.0103i)3-s + (−0.111 − 0.487i)4-s + (0.369 − 0.251i)5-s + (−0.0244 + 0.0423i)6-s + (−0.703 − 1.21i)7-s + (0.318 + 0.153i)8-s + (−0.950 + 0.293i)9-s + (−0.0236 + 0.315i)10-s + (0.0780 − 0.341i)11-s + (−0.0126 − 0.0322i)12-s + (0.0148 + 0.197i)13-s + (0.983 + 0.148i)14-s + (0.0226 − 0.0210i)15-s + (−0.225 + 0.108i)16-s + (−1.30 − 0.886i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.215 + 0.976i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.215 + 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(430\)    =    \(2 \cdot 5 \cdot 43\)
Sign: $-0.215 + 0.976i$
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: \((2,\ 430,\ (\ :1/2),\ -0.215 + 0.976i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.369267 - 0.459514i\)
\(L(\frac12)\) \(\approx\) \(0.369267 - 0.459514i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.31 + 6.13i)T \)
good3 \( 1 + (-0.118 + 0.0178i)T + (2.86 - 0.884i)T^{2} \)
7 \( 1 + (1.86 + 3.22i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.258 + 1.13i)T + (-9.91 - 4.77i)T^{2} \)
13 \( 1 + (-0.0534 - 0.713i)T + (-12.8 + 1.93i)T^{2} \)
17 \( 1 + (5.36 + 3.65i)T + (6.21 + 15.8i)T^{2} \)
19 \( 1 + (7.17 + 2.21i)T + (15.6 + 10.7i)T^{2} \)
23 \( 1 + (-0.285 - 0.265i)T + (1.71 + 22.9i)T^{2} \)
29 \( 1 + (-8.28 - 1.24i)T + (27.7 + 8.54i)T^{2} \)
31 \( 1 + (-0.767 - 1.95i)T + (-22.7 + 21.0i)T^{2} \)
37 \( 1 + (-2.32 + 4.02i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (2.77 - 3.48i)T + (-9.12 - 39.9i)T^{2} \)
47 \( 1 + (2.80 + 12.2i)T + (-42.3 + 20.3i)T^{2} \)
53 \( 1 + (-0.587 + 7.84i)T + (-52.4 - 7.89i)T^{2} \)
59 \( 1 + (-12.0 + 5.81i)T + (36.7 - 46.1i)T^{2} \)
61 \( 1 + (4.88 - 12.4i)T + (-44.7 - 41.4i)T^{2} \)
67 \( 1 + (-1.99 - 0.615i)T + (55.3 + 37.7i)T^{2} \)
71 \( 1 + (0.433 - 0.402i)T + (5.30 - 70.8i)T^{2} \)
73 \( 1 + (-0.868 - 11.5i)T + (-72.1 + 10.8i)T^{2} \)
79 \( 1 + (-2.93 - 5.08i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.386 - 0.0582i)T + (79.3 - 24.4i)T^{2} \)
89 \( 1 + (8.78 - 1.32i)T + (85.0 - 26.2i)T^{2} \)
97 \( 1 + (0.148 - 0.650i)T + (-87.3 - 42.0i)T^{2} \)
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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

−10.71145362646948057239746040080, −9.986384289367210845340965887971, −8.857813030385113745493271394739, −8.400022525819142094874754864139, −6.91292528338772385050269703202, −6.55963612203951652609010208411, −5.22028718806464443631583582834, −4.10848923539074819563063202957, −2.48727893965515693066797189763, −0.39861479630690654990243689899, 2.17286912149734135741484919839, 2.97317099397452236851697832809, 4.43957754590491345455067114368, 6.07108573035496429723360189590, 6.44185348704340474192286218157, 8.211803889040177954357890508312, 8.773888687606345370312382859213, 9.562779187252656563491883303954, 10.52158173311552260264074298014, 11.30957453695482931242127252771

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