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Properties

Label 2-430-5.4-c1-0-4
Degree $2$
Conductor $430$
Sign $-0.991 + 0.132i$
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 + 3.14i·3-s − 4-s + (2.21 − 0.296i)5-s − 3.14·6-s − 0.417i·7-s − i·8-s − 6.89·9-s + (0.296 + 2.21i)10-s − 2.87·11-s − 3.14i·12-s + 5.72i·13-s + 0.417·14-s + (0.932 + 6.97i)15-s + 16-s + 1.01i·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + 1.81i·3-s − 0.5·4-s + (0.991 − 0.132i)5-s − 1.28·6-s − 0.157i·7-s − 0.353i·8-s − 2.29·9-s + (0.0937 + 0.700i)10-s − 0.866·11-s − 0.908i·12-s + 1.58i·13-s + 0.111·14-s + (0.240 + 1.80i)15-s + 0.250·16-s + 0.244i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 + 0.132i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.991 + 0.132i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(430\)    =    \(2 \cdot 5 \cdot 43\)
Sign: $-0.991 + 0.132i$
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: \((2,\ 430,\ (\ :1/2),\ -0.991 + 0.132i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0846474 - 1.27145i\)
\(L(\frac12)\) \(\approx\) \(0.0846474 - 1.27145i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.21 + 0.296i)T \)
43 \( 1 + iT \)
good3 \( 1 - 3.14iT - 3T^{2} \)
7 \( 1 + 0.417iT - 7T^{2} \)
11 \( 1 + 2.87T + 11T^{2} \)
13 \( 1 - 5.72iT - 13T^{2} \)
17 \( 1 - 1.01iT - 17T^{2} \)
19 \( 1 + 6.84T + 19T^{2} \)
23 \( 1 + 0.439iT - 23T^{2} \)
29 \( 1 - 9.81T + 29T^{2} \)
31 \( 1 - 7.56T + 31T^{2} \)
37 \( 1 + 0.414iT - 37T^{2} \)
41 \( 1 + 0.907T + 41T^{2} \)
47 \( 1 - 7.50iT - 47T^{2} \)
53 \( 1 - 3.08iT - 53T^{2} \)
59 \( 1 - 2.57T + 59T^{2} \)
61 \( 1 - 8.18T + 61T^{2} \)
67 \( 1 - 7.29iT - 67T^{2} \)
71 \( 1 - 7.24T + 71T^{2} \)
73 \( 1 + 8.59iT - 73T^{2} \)
79 \( 1 + 13.9T + 79T^{2} \)
83 \( 1 - 4.13iT - 83T^{2} \)
89 \( 1 + 13.5T + 89T^{2} \)
97 \( 1 + 10.6iT - 97T^{2} \)
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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

−11.26789123962494801988686974857, −10.28116464212106922109298993973, −9.967472265752910473159215381673, −8.879639328416249395701798075475, −8.451025818276664411616186123427, −6.65978957550664421339799328965, −5.85655407110799807268650620020, −4.72347425235093750429429901410, −4.27799310964917768340391068734, −2.61221888959339582922535223307, 0.811598249034566926254680794974, 2.27979972250880020117757030335, 2.84576239255514464944657629023, 5.13410872448367214773855130745, 6.01992930659734789501834863710, 6.87423281342234783840531332419, 8.155719289813937822144366462810, 8.535584861598431130905131440989, 10.09930628974343705856653439789, 10.65763059937813108714223272266

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