Properties

Label 2-430-43.17-c1-0-3
Degree $2$
Conductor $430$
Sign $-0.686 - 0.726i$
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  + (0.222 + 0.974i)2-s + (0.485 − 0.149i)3-s + (−0.900 + 0.433i)4-s + (0.365 − 0.930i)5-s + (0.254 + 0.440i)6-s + (−2.29 + 3.96i)7-s + (−0.623 − 0.781i)8-s + (−2.26 + 1.54i)9-s + (0.988 + 0.149i)10-s + (2.13 + 1.03i)11-s + (−0.372 + 0.345i)12-s + (−2.32 + 0.351i)13-s + (−4.37 − 1.35i)14-s + (0.0379 − 0.506i)15-s + (0.623 − 0.781i)16-s + (1.59 + 4.07i)17-s + ⋯
L(s)  = 1  + (0.157 + 0.689i)2-s + (0.280 − 0.0865i)3-s + (−0.450 + 0.216i)4-s + (0.163 − 0.416i)5-s + (0.103 + 0.179i)6-s + (−0.865 + 1.49i)7-s + (−0.220 − 0.276i)8-s + (−0.755 + 0.514i)9-s + (0.312 + 0.0471i)10-s + (0.644 + 0.310i)11-s + (−0.107 + 0.0998i)12-s + (−0.645 + 0.0973i)13-s + (−1.16 − 0.360i)14-s + (0.00980 − 0.130i)15-s + (0.155 − 0.195i)16-s + (0.387 + 0.987i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.686 - 0.726i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.686 - 0.726i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(430\)    =    \(2 \cdot 5 \cdot 43\)
Sign: $-0.686 - 0.726i$
Analytic conductor: \(3.43356\)
Root analytic conductor: \(1.85298\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{430} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 430,\ (\ :1/2),\ -0.686 - 0.726i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.474111 + 1.10032i\)
\(L(\frac12)\) \(\approx\) \(0.474111 + 1.10032i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.222 - 0.974i)T \)
5 \( 1 + (-0.365 + 0.930i)T \)
43 \( 1 + (-6.39 - 1.45i)T \)
good3 \( 1 + (-0.485 + 0.149i)T + (2.47 - 1.68i)T^{2} \)
7 \( 1 + (2.29 - 3.96i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2.13 - 1.03i)T + (6.85 + 8.60i)T^{2} \)
13 \( 1 + (2.32 - 0.351i)T + (12.4 - 3.83i)T^{2} \)
17 \( 1 + (-1.59 - 4.07i)T + (-12.4 + 11.5i)T^{2} \)
19 \( 1 + (-0.837 - 0.570i)T + (6.94 + 17.6i)T^{2} \)
23 \( 1 + (-0.109 - 1.45i)T + (-22.7 + 3.42i)T^{2} \)
29 \( 1 + (-8.24 - 2.54i)T + (23.9 + 16.3i)T^{2} \)
31 \( 1 + (2.25 - 2.09i)T + (2.31 - 30.9i)T^{2} \)
37 \( 1 + (3.06 + 5.30i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (2.17 + 9.50i)T + (-36.9 + 17.7i)T^{2} \)
47 \( 1 + (-0.199 + 0.0959i)T + (29.3 - 36.7i)T^{2} \)
53 \( 1 + (10.3 + 1.56i)T + (50.6 + 15.6i)T^{2} \)
59 \( 1 + (-3.11 + 3.90i)T + (-13.1 - 57.5i)T^{2} \)
61 \( 1 + (-5.32 - 4.93i)T + (4.55 + 60.8i)T^{2} \)
67 \( 1 + (-8.06 - 5.49i)T + (24.4 + 62.3i)T^{2} \)
71 \( 1 + (0.374 - 4.99i)T + (-70.2 - 10.5i)T^{2} \)
73 \( 1 + (3.62 - 0.547i)T + (69.7 - 21.5i)T^{2} \)
79 \( 1 + (-7.44 + 12.8i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-10.5 + 3.25i)T + (68.5 - 46.7i)T^{2} \)
89 \( 1 + (-1.24 + 0.382i)T + (73.5 - 50.1i)T^{2} \)
97 \( 1 + (-13.9 - 6.72i)T + (60.4 + 75.8i)T^{2} \)
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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

−11.90836014162046336582397356669, −10.38926560703865027662541649295, −9.256549734541132432445452391885, −8.851368707121026039721703138066, −7.934015047911213415912742189268, −6.69210407123532174295441177935, −5.78491569608151075212758426657, −5.08247329876893658446986356332, −3.51244099533447199896367036314, −2.24376601647469655749928313581, 0.69907695523762808271444246057, 2.83857149455378393573542375942, 3.53000572005083702775001736048, 4.68754690434810935827856707442, 6.20454510830849773313092994204, 6.98710475253217028697317724678, 8.155816610645229304512008332736, 9.490930870663616842345496000691, 9.829000995346011195788795611275, 10.82011430285917218392534275804

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