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Properties

Label 2-430-215.108-c1-0-3
Degree $2$
Conductor $430$
Sign $-0.551 - 0.834i$
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  + (0.330 + 0.943i)2-s + (0.302 − 0.864i)3-s + (−0.781 + 0.623i)4-s + (−1.46 + 1.68i)5-s + 0.915·6-s + (−0.505 − 0.505i)7-s + (−0.846 − 0.532i)8-s + (1.69 + 1.34i)9-s + (−2.07 − 0.825i)10-s + (−0.780 + 0.978i)11-s + (0.302 + 0.864i)12-s + (−2.95 + 4.70i)13-s + (0.310 − 0.644i)14-s + (1.01 + 1.77i)15-s + (0.222 − 0.974i)16-s + (3.23 + 5.14i)17-s + ⋯
L(s)  = 1  + (0.233 + 0.667i)2-s + (0.174 − 0.498i)3-s + (−0.390 + 0.311i)4-s + (−0.655 + 0.755i)5-s + 0.373·6-s + (−0.191 − 0.191i)7-s + (−0.299 − 0.188i)8-s + (0.563 + 0.449i)9-s + (−0.657 − 0.261i)10-s + (−0.235 + 0.294i)11-s + (0.0872 + 0.249i)12-s + (−0.820 + 1.30i)13-s + (0.0829 − 0.172i)14-s + (0.262 + 0.458i)15-s + (0.0556 − 0.243i)16-s + (0.784 + 1.24i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.551 - 0.834i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.551 - 0.834i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(430\)    =    \(2 \cdot 5 \cdot 43\)
Sign: $-0.551 - 0.834i$
Analytic conductor: \(3.43356\)
Root analytic conductor: \(1.85298\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{430} (323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 430,\ (\ :1/2),\ -0.551 - 0.834i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.560053 + 1.04166i\)
\(L(\frac12)\) \(\approx\) \(0.560053 + 1.04166i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.330 - 0.943i)T \)
5 \( 1 + (1.46 - 1.68i)T \)
43 \( 1 + (3.36 + 5.62i)T \)
good3 \( 1 + (-0.302 + 0.864i)T + (-2.34 - 1.87i)T^{2} \)
7 \( 1 + (0.505 + 0.505i)T + 7iT^{2} \)
11 \( 1 + (0.780 - 0.978i)T + (-2.44 - 10.7i)T^{2} \)
13 \( 1 + (2.95 - 4.70i)T + (-5.64 - 11.7i)T^{2} \)
17 \( 1 + (-3.23 - 5.14i)T + (-7.37 + 15.3i)T^{2} \)
19 \( 1 + (-0.626 - 0.785i)T + (-4.22 + 18.5i)T^{2} \)
23 \( 1 + (6.02 - 0.678i)T + (22.4 - 5.11i)T^{2} \)
29 \( 1 + (-0.882 - 0.424i)T + (18.0 + 22.6i)T^{2} \)
31 \( 1 + (2.65 + 1.27i)T + (19.3 + 24.2i)T^{2} \)
37 \( 1 + (-5.50 - 5.50i)T + 37iT^{2} \)
41 \( 1 + (-3.81 - 1.83i)T + (25.5 + 32.0i)T^{2} \)
47 \( 1 + (-3.28 - 0.370i)T + (45.8 + 10.4i)T^{2} \)
53 \( 1 + (1.91 - 1.20i)T + (22.9 - 47.7i)T^{2} \)
59 \( 1 + (-10.0 - 2.29i)T + (53.1 + 25.5i)T^{2} \)
61 \( 1 + (-0.0620 - 0.128i)T + (-38.0 + 47.6i)T^{2} \)
67 \( 1 + (7.17 + 0.808i)T + (65.3 + 14.9i)T^{2} \)
71 \( 1 + (-10.6 + 8.46i)T + (15.7 - 69.2i)T^{2} \)
73 \( 1 + (0.152 + 0.0960i)T + (31.6 + 65.7i)T^{2} \)
79 \( 1 - 1.57iT - 79T^{2} \)
83 \( 1 + (3.62 + 1.26i)T + (64.8 + 51.7i)T^{2} \)
89 \( 1 + (-8.96 + 4.31i)T + (55.4 - 69.5i)T^{2} \)
97 \( 1 + (1.85 + 16.4i)T + (-94.5 + 21.5i)T^{2} \)
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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

−11.72645168369338905292831373429, −10.42012300510953540112728261679, −9.757535416549103171197222001852, −8.308677926203470746403055031712, −7.60702796396165490478363384363, −6.97865303611831455891861050297, −6.08061631451990855515260742157, −4.58582689935355284798871107937, −3.70972426206527473169953831327, −2.10908770541448348418542520910, 0.69787569723476074783399994774, 2.79819620116174092953646082648, 3.82019908091719986098579987478, 4.84246274614060093119190057201, 5.68017981176221590040411945359, 7.37161962928026539422095938902, 8.205681841670176049322453332480, 9.395207322923137470654496389321, 9.830410276095332255077440075257, 10.85261214672156679777382295901

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