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Properties

Label 2-430-5.4-c1-0-0
Degree $2$
Conductor $430$
Sign $-0.662 + 0.749i$
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  + i·2-s + 2.48i·3-s − 4-s + (−1.48 + 1.67i)5-s − 2.48·6-s − 0.193i·7-s − i·8-s − 3.15·9-s + (−1.67 − 1.48i)10-s − 3.67·11-s − 2.48i·12-s − 2.35i·13-s + 0.193·14-s + (−4.15 − 3.67i)15-s + 16-s + 5.44i·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + 1.43i·3-s − 0.5·4-s + (−0.662 + 0.749i)5-s − 1.01·6-s − 0.0733i·7-s − 0.353i·8-s − 1.05·9-s + (−0.529 − 0.468i)10-s − 1.10·11-s − 0.716i·12-s − 0.651i·13-s + 0.0518·14-s + (−1.07 − 0.948i)15-s + 0.250·16-s + 1.32i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.662 + 0.749i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.662 + 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(430\)    =    \(2 \cdot 5 \cdot 43\)
Sign: $-0.662 + 0.749i$
Analytic conductor: \(3.43356\)
Root analytic conductor: \(1.85298\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{430} (259, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 430,\ (\ :1/2),\ -0.662 + 0.749i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.320035 - 0.710185i\)
\(L(\frac12)\) \(\approx\) \(0.320035 - 0.710185i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
5 \( 1 + (1.48 - 1.67i)T \)
43 \( 1 - iT \)
good3 \( 1 - 2.48iT - 3T^{2} \)
7 \( 1 + 0.193iT - 7T^{2} \)
11 \( 1 + 3.67T + 11T^{2} \)
13 \( 1 + 2.35iT - 13T^{2} \)
17 \( 1 - 5.44iT - 17T^{2} \)
19 \( 1 - 5.80T + 19T^{2} \)
23 \( 1 - 0.324iT - 23T^{2} \)
29 \( 1 + 0.287T + 29T^{2} \)
31 \( 1 + 8.02T + 31T^{2} \)
37 \( 1 - 6.06iT - 37T^{2} \)
41 \( 1 + 2.22T + 41T^{2} \)
47 \( 1 + 7.28iT - 47T^{2} \)
53 \( 1 - 8.46iT - 53T^{2} \)
59 \( 1 + 9.73T + 59T^{2} \)
61 \( 1 - 1.79T + 61T^{2} \)
67 \( 1 - 3.97iT - 67T^{2} \)
71 \( 1 + 11.1T + 71T^{2} \)
73 \( 1 - 4.80iT - 73T^{2} \)
79 \( 1 + 2.44T + 79T^{2} \)
83 \( 1 - 17.2iT - 83T^{2} \)
89 \( 1 - 14.4T + 89T^{2} \)
97 \( 1 + 7.44iT - 97T^{2} \)
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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

−11.42257177118393706828440562512, −10.44630864294635288970286327309, −10.20610932098710155963065408699, −8.998697900759164875040422550351, −8.007915118765791704289666860172, −7.28261859699221729281147268459, −5.86262504894252141220708906027, −5.04704483221589622358828710383, −3.91566442145129655382118194903, −3.11379426852302089219060232602, 0.49564602043173311008570106138, 1.88561150356206937283539584660, 3.18237198796517343579781034719, 4.73174050959866337808584633621, 5.67635285852787839968425765659, 7.32746257241518406826287072138, 7.59393791572359922260702935102, 8.764578345953434793914968903600, 9.547709511612086597707420330457, 10.94288409521358207675385543909

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