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Properties

Label 2-430-43.36-c1-0-1
Degree $2$
Conductor $430$
Sign $-0.840 - 0.541i$
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 + (1 + 1.73i)3-s + 4-s + (−0.5 − 0.866i)5-s + (−1 − 1.73i)6-s + (−1.58 + 2.73i)7-s − 8-s + (−0.499 + 0.866i)9-s + (0.5 + 0.866i)10-s − 4.16·11-s + (1 + 1.73i)12-s + (−1.58 + 2.73i)13-s + (1.58 − 2.73i)14-s + (0.999 − 1.73i)15-s + 16-s + (−2.08 + 3.60i)17-s + ⋯
L(s)  = 1  − 0.707·2-s + (0.577 + 0.999i)3-s + 0.5·4-s + (−0.223 − 0.387i)5-s + (−0.408 − 0.707i)6-s + (−0.597 + 1.03i)7-s − 0.353·8-s + (−0.166 + 0.288i)9-s + (0.158 + 0.273i)10-s − 1.25·11-s + (0.288 + 0.499i)12-s + (−0.438 + 0.759i)13-s + (0.422 − 0.731i)14-s + (0.258 − 0.447i)15-s + 0.250·16-s + (−0.504 + 0.874i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.840 - 0.541i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.840 - 0.541i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(430\)    =    \(2 \cdot 5 \cdot 43\)
Sign: $-0.840 - 0.541i$
Analytic conductor: \(3.43356\)
Root analytic conductor: \(1.85298\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{430} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 430,\ (\ :1/2),\ -0.840 - 0.541i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.205304 + 0.697880i\)
\(L(\frac12)\) \(\approx\) \(0.205304 + 0.697880i\)
\(L(\frac{3}{2})\) 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 \( 1 + (4.66 - 4.61i)T \)
good3 \( 1 + (-1 - 1.73i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 + (1.58 - 2.73i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + 4.16T + 11T^{2} \)
13 \( 1 + (1.58 - 2.73i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (2.08 - 3.60i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.08 - 1.87i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.16 + 2.01i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-1.41 - 2.45i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (1 + 1.73i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 11.3T + 41T^{2} \)
47 \( 1 - 8.32T + 47T^{2} \)
53 \( 1 + (1.16 + 2.01i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 10.1T + 59T^{2} \)
61 \( 1 + (-2.58 + 4.47i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.82 - 10.0i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (2.41 - 4.18i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (3.08 - 5.33i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (1.58 - 2.73i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (3 + 5.19i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (1.5 + 2.59i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 3.83T + 97T^{2} \)
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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

−11.37324352629293239862044261879, −10.13737606544185944030347315396, −9.834636546562734706266632764600, −8.683374437777238674227538440298, −8.455045779018717607091245123872, −7.05148519318817135856948917332, −5.81478389051007572530365409793, −4.69091655557659445931945952318, −3.40227139491276951917314056207, −2.27076426258596303575280060471, 0.50871132316289690088736390091, 2.35669848090057977057481462836, 3.25925852659407306975816869845, 5.05964591032835850956600539635, 6.65885077754414738031865719978, 7.29364182955520077699804451527, 7.81582190248355035774060727475, 8.775274224634950694170776207758, 10.11833399155694018340359010532, 10.45758527402262492157987129064

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