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Properties

Label 2-430-43.6-c1-0-9
Degree $2$
Conductor $430$
Sign $0.434 + 0.900i$
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  − 2-s + (0.622 − 1.07i)3-s + 4-s + (0.5 − 0.866i)5-s + (−0.622 + 1.07i)6-s + (−0.561 − 0.972i)7-s − 8-s + (0.724 + 1.25i)9-s + (−0.5 + 0.866i)10-s + 4.76·11-s + (0.622 − 1.07i)12-s + (0.285 + 0.494i)13-s + (0.561 + 0.972i)14-s + (−0.622 − 1.07i)15-s + 16-s + (−2.28 − 3.95i)17-s + ⋯
L(s)  = 1  − 0.707·2-s + (0.359 − 0.622i)3-s + 0.5·4-s + (0.223 − 0.387i)5-s + (−0.254 + 0.440i)6-s + (−0.212 − 0.367i)7-s − 0.353·8-s + (0.241 + 0.418i)9-s + (−0.158 + 0.273i)10-s + 1.43·11-s + (0.179 − 0.311i)12-s + (0.0791 + 0.137i)13-s + (0.150 + 0.259i)14-s + (−0.160 − 0.278i)15-s + 0.250·16-s + (−0.554 − 0.960i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.434 + 0.900i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.434 + 0.900i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(430\)    =    \(2 \cdot 5 \cdot 43\)
Sign: $0.434 + 0.900i$
Analytic conductor: \(3.43356\)
Root analytic conductor: \(1.85298\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{430} (221, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 430,\ (\ :1/2),\ 0.434 + 0.900i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.04659 - 0.656789i\)
\(L(\frac12)\) \(\approx\) \(1.04659 - 0.656789i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
5 \( 1 + (-0.5 + 0.866i)T \)
43 \( 1 + (-5.14 + 4.06i)T \)
good3 \( 1 + (-0.622 + 1.07i)T + (-1.5 - 2.59i)T^{2} \)
7 \( 1 + (0.561 + 0.972i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 - 4.76T + 11T^{2} \)
13 \( 1 + (-0.285 - 0.494i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (2.28 + 3.95i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.63 + 6.28i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.06 - 5.31i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.44 + 5.96i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.23 - 5.59i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.90 + 3.29i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 1.49T + 41T^{2} \)
47 \( 1 + 9.44T + 47T^{2} \)
53 \( 1 + (-2.52 + 4.36i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 4.57T + 59T^{2} \)
61 \( 1 + (-5.90 - 10.2i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.16 + 5.48i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-5.53 - 9.59i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-3.43 - 5.95i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-6.12 - 10.6i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (8.89 - 15.4i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (6.88 - 11.9i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 12.1T + 97T^{2} \)
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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

−11.12442570527740921163143468751, −9.701897289611031228713073833114, −9.297392388726706982783510451194, −8.315978019913136532858109389508, −7.18858956284738698463278467765, −6.81864141556985787400379604275, −5.34931888819240640970336719107, −3.93671600808891888770418719721, −2.35126775199540579799462819000, −1.09050467790173372905873400962, 1.66048115017650996643822356899, 3.29173891971823920332315718318, 4.16921213297448516075449617112, 5.98055788844113732838103743463, 6.57315146781947928094417887144, 7.84192307361827176146442256533, 8.879426489468409492483069529661, 9.479048464698940821406023588515, 10.18381681692228225756081594333, 11.09077714595621114971295032142

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