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Properties

Label 2-430-43.25-c1-0-6
Degree $2$
Conductor $430$
Sign $0.175 - 0.984i$
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  + (0.900 + 0.433i)2-s + (1.41 + 0.967i)3-s + (0.623 + 0.781i)4-s + (−0.733 + 0.680i)5-s + (0.858 + 1.48i)6-s + (−0.475 + 0.823i)7-s + (0.222 + 0.974i)8-s + (−0.0189 − 0.0482i)9-s + (−0.955 + 0.294i)10-s + (−1.16 + 1.45i)11-s + (0.128 + 1.71i)12-s + (5.50 + 1.69i)13-s + (−0.785 + 0.535i)14-s + (−1.69 + 0.255i)15-s + (−0.222 + 0.974i)16-s + (0.502 + 0.466i)17-s + ⋯
L(s)  = 1  + (0.637 + 0.306i)2-s + (0.819 + 0.558i)3-s + (0.311 + 0.390i)4-s + (−0.327 + 0.304i)5-s + (0.350 + 0.607i)6-s + (−0.179 + 0.311i)7-s + (0.0786 + 0.344i)8-s + (−0.00630 − 0.0160i)9-s + (−0.302 + 0.0932i)10-s + (−0.350 + 0.439i)11-s + (0.0370 + 0.494i)12-s + (1.52 + 0.470i)13-s + (−0.210 + 0.143i)14-s + (−0.438 + 0.0660i)15-s + (−0.0556 + 0.243i)16-s + (0.121 + 0.113i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.175 - 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.175 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(430\)    =    \(2 \cdot 5 \cdot 43\)
Sign: $0.175 - 0.984i$
Analytic conductor: \(3.43356\)
Root analytic conductor: \(1.85298\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{430} (111, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 430,\ (\ :1/2),\ 0.175 - 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.82379 + 1.52805i\)
\(L(\frac12)\) \(\approx\) \(1.82379 + 1.52805i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.900 - 0.433i)T \)
5 \( 1 + (0.733 - 0.680i)T \)
43 \( 1 + (-6.29 + 1.82i)T \)
good3 \( 1 + (-1.41 - 0.967i)T + (1.09 + 2.79i)T^{2} \)
7 \( 1 + (0.475 - 0.823i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.16 - 1.45i)T + (-2.44 - 10.7i)T^{2} \)
13 \( 1 + (-5.50 - 1.69i)T + (10.7 + 7.32i)T^{2} \)
17 \( 1 + (-0.502 - 0.466i)T + (1.27 + 16.9i)T^{2} \)
19 \( 1 + (0.423 - 1.08i)T + (-13.9 - 12.9i)T^{2} \)
23 \( 1 + (2.70 + 0.408i)T + (21.9 + 6.77i)T^{2} \)
29 \( 1 + (-0.948 + 0.646i)T + (10.5 - 26.9i)T^{2} \)
31 \( 1 + (0.658 + 8.78i)T + (-30.6 + 4.62i)T^{2} \)
37 \( 1 + (2.53 + 4.38i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (3.09 + 1.48i)T + (25.5 + 32.0i)T^{2} \)
47 \( 1 + (2.95 + 3.70i)T + (-10.4 + 45.8i)T^{2} \)
53 \( 1 + (-3.17 + 0.979i)T + (43.7 - 29.8i)T^{2} \)
59 \( 1 + (-1.43 + 6.29i)T + (-53.1 - 25.5i)T^{2} \)
61 \( 1 + (0.0511 - 0.682i)T + (-60.3 - 9.09i)T^{2} \)
67 \( 1 + (-0.390 + 0.995i)T + (-49.1 - 45.5i)T^{2} \)
71 \( 1 + (2.22 - 0.335i)T + (67.8 - 20.9i)T^{2} \)
73 \( 1 + (-0.0726 - 0.0224i)T + (60.3 + 41.1i)T^{2} \)
79 \( 1 + (5.92 - 10.2i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-4.78 - 3.26i)T + (30.3 + 77.2i)T^{2} \)
89 \( 1 + (-7.28 - 4.96i)T + (32.5 + 82.8i)T^{2} \)
97 \( 1 + (-6.39 + 8.02i)T + (-21.5 - 94.5i)T^{2} \)
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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

−11.43484114338332122300092821273, −10.47593474113766644305242720955, −9.427472955162403956186221426097, −8.567260904363302699920592914605, −7.78080427223229688413438368316, −6.55675885038915129988098132969, −5.69651167023415888414181547890, −4.17000280536737754322823475683, −3.62165091153492120005344435060, −2.35955807346117412971114917778, 1.34872465771680356971116082826, 2.90571334421493873411393737370, 3.72985649623845147931712912222, 5.08388455754942394995881730120, 6.19672484878972246729556558713, 7.31106602359490376156844656286, 8.279180409148534997673560137290, 8.868365942055372822636013688923, 10.32252503433627069835086323633, 10.99023041834033326712749054588

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