Properties

Label 2-430-43.6-c1-0-10
Degree $2$
Conductor $430$
Sign $-0.164 + 0.986i$
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 + (−0.850 + 1.47i)3-s + 4-s + (0.5 − 0.866i)5-s + (0.850 − 1.47i)6-s + (−1.11 − 1.92i)7-s − 8-s + (0.0518 + 0.0897i)9-s + (−0.5 + 0.866i)10-s − 2.26·11-s + (−0.850 + 1.47i)12-s + (−2.41 − 4.17i)13-s + (1.11 + 1.92i)14-s + (0.850 + 1.47i)15-s + 16-s + (−2.16 − 3.74i)17-s + ⋯
L(s)  = 1  − 0.707·2-s + (−0.491 + 0.850i)3-s + 0.5·4-s + (0.223 − 0.387i)5-s + (0.347 − 0.601i)6-s + (−0.420 − 0.728i)7-s − 0.353·8-s + (0.0172 + 0.0299i)9-s + (−0.158 + 0.273i)10-s − 0.682·11-s + (−0.245 + 0.425i)12-s + (−0.668 − 1.15i)13-s + (0.297 + 0.515i)14-s + (0.219 + 0.380i)15-s + 0.250·16-s + (−0.524 − 0.909i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.164 + 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.164 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(430\)    =    \(2 \cdot 5 \cdot 43\)
Sign: $-0.164 + 0.986i$
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: \((2,\ 430,\ (\ :1/2),\ -0.164 + 0.986i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.267206 - 0.315569i\)
\(L(\frac12)\) \(\approx\) \(0.267206 - 0.315569i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-6.54 + 0.347i)T \)
good3 \( 1 + (0.850 - 1.47i)T + (-1.5 - 2.59i)T^{2} \)
7 \( 1 + (1.11 + 1.92i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + 2.26T + 11T^{2} \)
13 \( 1 + (2.41 + 4.17i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (2.16 + 3.74i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.83 - 4.90i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.37 + 2.37i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.480 + 0.832i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-5.00 + 8.67i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-5.29 + 9.17i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 4.40T + 41T^{2} \)
47 \( 1 + 10.0T + 47T^{2} \)
53 \( 1 + (1.56 - 2.70i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 4.32T + 59T^{2} \)
61 \( 1 + (5.97 + 10.3i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.62 - 9.73i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-6.74 - 11.6i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-5.46 - 9.46i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (3.66 + 6.34i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (4.58 - 7.94i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-4.94 + 8.55i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 2.19T + 97T^{2} \)
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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

−10.59770400998235406489057885179, −10.05804141632302886491842433879, −9.462660000058314433268766964893, −8.127836914677526574660373235193, −7.43953410826454967668880524596, −6.10256208170445444424522370522, −5.13368963802008694996558658795, −4.09437303659550858809227164301, −2.51238376828915303457380890989, −0.33263100803694561495496252114, 1.74340267477197427036073076928, 2.89004860172447588583435172029, 4.82373005547025857914312041582, 6.32178261014652734486696983406, 6.59160030355496190590035219143, 7.62335560745036358275494829657, 8.775207487518744406559669737446, 9.518040182725539196665962419121, 10.53373952876036574142706840359, 11.42553031375415881811485979819

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