Properties

Label 2-430-43.41-c1-0-4
Degree $2$
Conductor $430$
Sign $0.959 + 0.282i$
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 + (0.403 + 0.194i)3-s + (0.623 − 0.781i)4-s + (−0.222 + 0.974i)5-s + 0.447·6-s + 2.78·7-s + (0.222 − 0.974i)8-s + (−1.74 − 2.18i)9-s + (0.222 + 0.974i)10-s + (3.02 + 3.79i)11-s + (0.403 − 0.194i)12-s + (0.482 − 2.11i)13-s + (2.50 − 1.20i)14-s + (−0.278 + 0.349i)15-s + (−0.222 − 0.974i)16-s + (0.129 + 0.568i)17-s + ⋯
L(s)  = 1  + (0.637 − 0.306i)2-s + (0.232 + 0.112i)3-s + (0.311 − 0.390i)4-s + (−0.0995 + 0.436i)5-s + 0.182·6-s + 1.05·7-s + (0.0786 − 0.344i)8-s + (−0.581 − 0.729i)9-s + (0.0703 + 0.308i)10-s + (0.912 + 1.14i)11-s + (0.116 − 0.0560i)12-s + (0.133 − 0.585i)13-s + (0.669 − 0.322i)14-s + (−0.0720 + 0.0903i)15-s + (−0.0556 − 0.243i)16-s + (0.0314 + 0.137i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.959 + 0.282i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.959 + 0.282i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(430\)    =    \(2 \cdot 5 \cdot 43\)
Sign: $0.959 + 0.282i$
Analytic conductor: \(3.43356\)
Root analytic conductor: \(1.85298\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{430} (41, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 430,\ (\ :1/2),\ 0.959 + 0.282i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.26679 - 0.326837i\)
\(L(\frac12)\) \(\approx\) \(2.26679 - 0.326837i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.222 - 0.974i)T \)
43 \( 1 + (3.73 + 5.39i)T \)
good3 \( 1 + (-0.403 - 0.194i)T + (1.87 + 2.34i)T^{2} \)
7 \( 1 - 2.78T + 7T^{2} \)
11 \( 1 + (-3.02 - 3.79i)T + (-2.44 + 10.7i)T^{2} \)
13 \( 1 + (-0.482 + 2.11i)T + (-11.7 - 5.64i)T^{2} \)
17 \( 1 + (-0.129 - 0.568i)T + (-15.3 + 7.37i)T^{2} \)
19 \( 1 + (-0.315 + 0.395i)T + (-4.22 - 18.5i)T^{2} \)
23 \( 1 + (-1.20 - 1.51i)T + (-5.11 + 22.4i)T^{2} \)
29 \( 1 + (-2.41 + 1.16i)T + (18.0 - 22.6i)T^{2} \)
31 \( 1 + (2.05 - 0.991i)T + (19.3 - 24.2i)T^{2} \)
37 \( 1 + 4.50T + 37T^{2} \)
41 \( 1 + (3.01 - 1.44i)T + (25.5 - 32.0i)T^{2} \)
47 \( 1 + (3.13 - 3.92i)T + (-10.4 - 45.8i)T^{2} \)
53 \( 1 + (-1.64 - 7.22i)T + (-47.7 + 22.9i)T^{2} \)
59 \( 1 + (2.67 + 11.7i)T + (-53.1 + 25.5i)T^{2} \)
61 \( 1 + (7.37 + 3.55i)T + (38.0 + 47.6i)T^{2} \)
67 \( 1 + (4.32 - 5.41i)T + (-14.9 - 65.3i)T^{2} \)
71 \( 1 + (0.495 - 0.620i)T + (-15.7 - 69.2i)T^{2} \)
73 \( 1 + (2.26 - 9.93i)T + (-65.7 - 31.6i)T^{2} \)
79 \( 1 + 8.10T + 79T^{2} \)
83 \( 1 + (-2.29 - 1.10i)T + (51.7 + 64.8i)T^{2} \)
89 \( 1 + (1.65 + 0.798i)T + (55.4 + 69.5i)T^{2} \)
97 \( 1 + (-7.43 - 9.32i)T + (-21.5 + 94.5i)T^{2} \)
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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

−11.33242096108068296819016136517, −10.36322303831040276973595974761, −9.422717977874732828074644209352, −8.404764848998439009700007941712, −7.30112310817344580811415038128, −6.35473866466512059475819110965, −5.18557867761859552732447920578, −4.15483247743974702528498944621, −3.10012455052270574576393611690, −1.65720463114967979849387776673, 1.68636325223092348657991607177, 3.25685016337722120482121162316, 4.49965128838853108542861981235, 5.34517527615400048229222540266, 6.38226845894829702987707368392, 7.57476668366788036860235370979, 8.474247125813890444494210583132, 8.945690991963120133432493858172, 10.59815108828133884583740802097, 11.50116506511089428729710069761

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