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Properties

Label 2-430-5.4-c1-0-10
Degree $2$
Conductor $430$
Sign $0.970 - 0.241i$
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 + 1.17i·3-s − 4-s + (2.17 − 0.539i)5-s + 1.17·6-s + 2.70i·7-s + i·8-s + 1.63·9-s + (−0.539 − 2.17i)10-s − 2.53·11-s − 1.17i·12-s + 0.0783i·13-s + 2.70·14-s + (0.630 + 2.53i)15-s + 16-s + 5.51i·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + 0.675i·3-s − 0.5·4-s + (0.970 − 0.241i)5-s + 0.477·6-s + 1.02i·7-s + 0.353i·8-s + 0.543·9-s + (−0.170 − 0.686i)10-s − 0.765·11-s − 0.337i·12-s + 0.0217i·13-s + 0.724·14-s + (0.162 + 0.655i)15-s + 0.250·16-s + 1.33i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 - 0.241i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.970 - 0.241i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(430\)    =    \(2 \cdot 5 \cdot 43\)
Sign: $0.970 - 0.241i$
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: \((2,\ 430,\ (\ :1/2),\ 0.970 - 0.241i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.54670 + 0.189273i\)
\(L(\frac12)\) \(\approx\) \(1.54670 + 0.189273i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.17 + 0.539i)T \)
43 \( 1 + iT \)
good3 \( 1 - 1.17iT - 3T^{2} \)
7 \( 1 - 2.70iT - 7T^{2} \)
11 \( 1 + 2.53T + 11T^{2} \)
13 \( 1 - 0.0783iT - 13T^{2} \)
17 \( 1 - 5.51iT - 17T^{2} \)
19 \( 1 - 3.29T + 19T^{2} \)
23 \( 1 + 1.46iT - 23T^{2} \)
29 \( 1 - 5.87T + 29T^{2} \)
31 \( 1 + 4.61T + 31T^{2} \)
37 \( 1 + 9.95iT - 37T^{2} \)
41 \( 1 - 7.83T + 41T^{2} \)
47 \( 1 - 1.12iT - 47T^{2} \)
53 \( 1 - 5.89iT - 53T^{2} \)
59 \( 1 + 12.4T + 59T^{2} \)
61 \( 1 + 11.4T + 61T^{2} \)
67 \( 1 + 7.38iT - 67T^{2} \)
71 \( 1 + 2.98T + 71T^{2} \)
73 \( 1 + 11.4iT - 73T^{2} \)
79 \( 1 - 8.51T + 79T^{2} \)
83 \( 1 + 12.9iT - 83T^{2} \)
89 \( 1 + 16.3T + 89T^{2} \)
97 \( 1 + 3.51iT - 97T^{2} \)
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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

−10.77620215401107245587972358560, −10.46965569183145206352013947300, −9.374301264920100378365838308284, −8.991453365135146115845163440416, −7.74605914411293993459521684999, −6.09842149639590370898045694963, −5.33331422339612600212546287453, −4.33903469775137551110217113012, −2.92068540425699385286658594211, −1.75021217101944983312230311255, 1.15821843740373829515087899277, 2.87340324050520458732078610333, 4.54639417093841164204066079301, 5.53232288033794923834525413611, 6.71041898973104206889542717402, 7.23653756626919484119946423116, 8.035686237582303125592954369832, 9.450111461142839547266228165075, 10.02478836152437559170969444529, 10.95516928017714548313388118674

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