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Properties

Label 2-430-43.35-c1-0-0
Degree $2$
Conductor $430$
Sign $-0.914 + 0.403i$
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 + (0.363 + 1.59i)3-s + (−0.900 − 0.433i)4-s + (−0.623 + 0.781i)5-s − 1.63·6-s − 1.57·7-s + (0.623 − 0.781i)8-s + (0.299 − 0.144i)9-s + (−0.623 − 0.781i)10-s + (−3.64 + 1.75i)11-s + (0.363 − 1.59i)12-s + (−2.75 + 3.45i)13-s + (0.351 − 1.53i)14-s + (−1.47 − 0.708i)15-s + (0.623 + 0.781i)16-s + (−0.0798 − 0.100i)17-s + ⋯
L(s)  = 1  + (−0.157 + 0.689i)2-s + (0.209 + 0.919i)3-s + (−0.450 − 0.216i)4-s + (−0.278 + 0.349i)5-s − 0.666·6-s − 0.597·7-s + (0.220 − 0.276i)8-s + (0.0998 − 0.0481i)9-s + (−0.197 − 0.247i)10-s + (−1.09 + 0.529i)11-s + (0.104 − 0.459i)12-s + (−0.763 + 0.957i)13-s + (0.0939 − 0.411i)14-s + (−0.379 − 0.182i)15-s + (0.155 + 0.195i)16-s + (−0.0193 − 0.0242i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.914 + 0.403i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.914 + 0.403i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(430\)    =    \(2 \cdot 5 \cdot 43\)
Sign: $-0.914 + 0.403i$
Analytic conductor: \(3.43356\)
Root analytic conductor: \(1.85298\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{430} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 430,\ (\ :1/2),\ -0.914 + 0.403i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.144197 - 0.684099i\)
\(L(\frac12)\) \(\approx\) \(0.144197 - 0.684099i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.623 - 0.781i)T \)
43 \( 1 + (5.90 - 2.84i)T \)
good3 \( 1 + (-0.363 - 1.59i)T + (-2.70 + 1.30i)T^{2} \)
7 \( 1 + 1.57T + 7T^{2} \)
11 \( 1 + (3.64 - 1.75i)T + (6.85 - 8.60i)T^{2} \)
13 \( 1 + (2.75 - 3.45i)T + (-2.89 - 12.6i)T^{2} \)
17 \( 1 + (0.0798 + 0.100i)T + (-3.78 + 16.5i)T^{2} \)
19 \( 1 + (3.10 + 1.49i)T + (11.8 + 14.8i)T^{2} \)
23 \( 1 + (2.67 - 1.28i)T + (14.3 - 17.9i)T^{2} \)
29 \( 1 + (0.120 - 0.529i)T + (-26.1 - 12.5i)T^{2} \)
31 \( 1 + (-1.42 + 6.26i)T + (-27.9 - 13.4i)T^{2} \)
37 \( 1 - 9.65T + 37T^{2} \)
41 \( 1 + (1.01 - 4.42i)T + (-36.9 - 17.7i)T^{2} \)
47 \( 1 + (-5.95 - 2.86i)T + (29.3 + 36.7i)T^{2} \)
53 \( 1 + (-7.69 - 9.64i)T + (-11.7 + 51.6i)T^{2} \)
59 \( 1 + (0.649 + 0.814i)T + (-13.1 + 57.5i)T^{2} \)
61 \( 1 + (-0.372 - 1.63i)T + (-54.9 + 26.4i)T^{2} \)
67 \( 1 + (4.12 + 1.98i)T + (41.7 + 52.3i)T^{2} \)
71 \( 1 + (-0.741 - 0.356i)T + (44.2 + 55.5i)T^{2} \)
73 \( 1 + (6.79 - 8.51i)T + (-16.2 - 71.1i)T^{2} \)
79 \( 1 + 7.44T + 79T^{2} \)
83 \( 1 + (-0.532 - 2.33i)T + (-74.7 + 36.0i)T^{2} \)
89 \( 1 + (-3.45 - 15.1i)T + (-80.1 + 38.6i)T^{2} \)
97 \( 1 + (-10.4 + 5.02i)T + (60.4 - 75.8i)T^{2} \)
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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

−11.51607056639142735204652938959, −10.38712523075467541125378078917, −9.804046251657291327710049722627, −9.139944908651719610374572700837, −7.914203175091260320661938060537, −7.10489223851897571978603145222, −6.11841648668531559688350099438, −4.75876104400639864499310730674, −4.09799883984725678173803040396, −2.61926157095278571870518855055, 0.43686922400753807065797429533, 2.17163961337280783033139952167, 3.22638202631560079801528984825, 4.68195586265276707835160925426, 5.87546108508662455294937258060, 7.17833603974120115536677840021, 8.011989207218532020173004356763, 8.640571415558442269460112357799, 10.06160071296161402777653912988, 10.45296989525959054634182314620

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