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Properties

Label 2-430-43.38-c1-0-1
Degree $2$
Conductor $430$
Sign $-0.621 - 0.783i$
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 + (−1.77 − 0.548i)3-s + (−0.900 − 0.433i)4-s + (−0.365 − 0.930i)5-s + (0.931 − 1.61i)6-s + (1.00 + 1.73i)7-s + (0.623 − 0.781i)8-s + (0.385 + 0.263i)9-s + (0.988 − 0.149i)10-s + (0.199 − 0.0958i)11-s + (1.36 + 1.26i)12-s + (1.92 + 0.290i)13-s + (−1.91 + 0.591i)14-s + (0.139 + 1.85i)15-s + (0.623 + 0.781i)16-s + (−1.22 + 3.13i)17-s + ⋯
L(s)  = 1  + (−0.157 + 0.689i)2-s + (−1.02 − 0.316i)3-s + (−0.450 − 0.216i)4-s + (−0.163 − 0.416i)5-s + (0.380 − 0.658i)6-s + (0.379 + 0.657i)7-s + (0.220 − 0.276i)8-s + (0.128 + 0.0877i)9-s + (0.312 − 0.0471i)10-s + (0.0600 − 0.0289i)11-s + (0.394 + 0.365i)12-s + (0.535 + 0.0806i)13-s + (−0.512 + 0.158i)14-s + (0.0359 + 0.479i)15-s + (0.155 + 0.195i)16-s + (−0.298 + 0.759i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.621 - 0.783i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.621 - 0.783i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(430\)    =    \(2 \cdot 5 \cdot 43\)
Sign: $-0.621 - 0.783i$
Analytic conductor: \(3.43356\)
Root analytic conductor: \(1.85298\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{430} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 430,\ (\ :1/2),\ -0.621 - 0.783i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.239564 + 0.495989i\)
\(L(\frac12)\) \(\approx\) \(0.239564 + 0.495989i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.532 - 6.53i)T \)
good3 \( 1 + (1.77 + 0.548i)T + (2.47 + 1.68i)T^{2} \)
7 \( 1 + (-1.00 - 1.73i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.199 + 0.0958i)T + (6.85 - 8.60i)T^{2} \)
13 \( 1 + (-1.92 - 0.290i)T + (12.4 + 3.83i)T^{2} \)
17 \( 1 + (1.22 - 3.13i)T + (-12.4 - 11.5i)T^{2} \)
19 \( 1 + (4.16 - 2.83i)T + (6.94 - 17.6i)T^{2} \)
23 \( 1 + (0.561 - 7.48i)T + (-22.7 - 3.42i)T^{2} \)
29 \( 1 + (6.89 - 2.12i)T + (23.9 - 16.3i)T^{2} \)
31 \( 1 + (-2.10 - 1.95i)T + (2.31 + 30.9i)T^{2} \)
37 \( 1 + (0.494 - 0.856i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.0413 + 0.181i)T + (-36.9 - 17.7i)T^{2} \)
47 \( 1 + (-4.59 - 2.21i)T + (29.3 + 36.7i)T^{2} \)
53 \( 1 + (3.09 - 0.466i)T + (50.6 - 15.6i)T^{2} \)
59 \( 1 + (5.30 + 6.65i)T + (-13.1 + 57.5i)T^{2} \)
61 \( 1 + (-0.213 + 0.197i)T + (4.55 - 60.8i)T^{2} \)
67 \( 1 + (-3.70 + 2.52i)T + (24.4 - 62.3i)T^{2} \)
71 \( 1 + (-0.0301 - 0.402i)T + (-70.2 + 10.5i)T^{2} \)
73 \( 1 + (-8.36 - 1.26i)T + (69.7 + 21.5i)T^{2} \)
79 \( 1 + (-8.79 - 15.2i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (5.34 + 1.64i)T + (68.5 + 46.7i)T^{2} \)
89 \( 1 + (3.50 + 1.08i)T + (73.5 + 50.1i)T^{2} \)
97 \( 1 + (-3.20 + 1.54i)T + (60.4 - 75.8i)T^{2} \)
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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

−11.43347258159230692459019496399, −10.80772228570856787399115105209, −9.483617879630014747293286120333, −8.614949624622218479157780087072, −7.82909742006538207512577649730, −6.59089920514625705612669730975, −5.84674636047305335547275665255, −5.15862529251582440495152902614, −3.83762083552004931573929461528, −1.54660920299840792663887683810, 0.43566263029986996754738299293, 2.43411239199973755387635613106, 4.01679516959179172636643261385, 4.79202089548496304621012157136, 6.02441663644293381191288155418, 7.04145609852168780397672455441, 8.197807021398981453741142526227, 9.214631287417097715184351571106, 10.48663124768728778068472522061, 10.77408532181930133951268896563

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