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Properties

Label 2-430-215.128-c1-0-18
Degree $2$
Conductor $430$
Sign $-0.997 - 0.0700i$
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  + (−0.707 − 0.707i)2-s + (−0.515 + 0.515i)3-s + 1.00i·4-s + (0.417 − 2.19i)5-s + 0.729·6-s + (−1.42 − 1.42i)7-s + (0.707 − 0.707i)8-s + 2.46i·9-s + (−1.84 + 1.25i)10-s − 4.26·11-s + (−0.515 − 0.515i)12-s + (−1.59 − 1.59i)13-s + 2.01i·14-s + (0.917 + 1.34i)15-s − 1.00·16-s + (2.57 − 2.57i)17-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + (−0.297 + 0.297i)3-s + 0.500i·4-s + (0.186 − 0.982i)5-s + 0.297·6-s + (−0.538 − 0.538i)7-s + (0.250 − 0.250i)8-s + 0.822i·9-s + (−0.584 + 0.397i)10-s − 1.28·11-s + (−0.148 − 0.148i)12-s + (−0.443 − 0.443i)13-s + 0.538i·14-s + (0.236 + 0.348i)15-s − 0.250·16-s + (0.624 − 0.624i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0700i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.997 - 0.0700i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(430\)    =    \(2 \cdot 5 \cdot 43\)
Sign: $-0.997 - 0.0700i$
Analytic conductor: \(3.43356\)
Root analytic conductor: \(1.85298\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{430} (343, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 430,\ (\ :1/2),\ -0.997 - 0.0700i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00973311 + 0.277385i\)
\(L(\frac12)\) \(\approx\) \(0.00973311 + 0.277385i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 + 0.707i)T \)
5 \( 1 + (-0.417 + 2.19i)T \)
43 \( 1 + (6.51 + 0.745i)T \)
good3 \( 1 + (0.515 - 0.515i)T - 3iT^{2} \)
7 \( 1 + (1.42 + 1.42i)T + 7iT^{2} \)
11 \( 1 + 4.26T + 11T^{2} \)
13 \( 1 + (1.59 + 1.59i)T + 13iT^{2} \)
17 \( 1 + (-2.57 + 2.57i)T - 17iT^{2} \)
19 \( 1 + 2.66T + 19T^{2} \)
23 \( 1 + (-3.44 - 3.44i)T + 23iT^{2} \)
29 \( 1 + 10.1T + 29T^{2} \)
31 \( 1 + 7.93T + 31T^{2} \)
37 \( 1 + (3.75 + 3.75i)T + 37iT^{2} \)
41 \( 1 - 10.3T + 41T^{2} \)
47 \( 1 + (3.90 - 3.90i)T - 47iT^{2} \)
53 \( 1 + (-0.248 - 0.248i)T + 53iT^{2} \)
59 \( 1 + 7.77iT - 59T^{2} \)
61 \( 1 - 11.7iT - 61T^{2} \)
67 \( 1 + (-9.53 + 9.53i)T - 67iT^{2} \)
71 \( 1 + 2.36iT - 71T^{2} \)
73 \( 1 + (-6.02 + 6.02i)T - 73iT^{2} \)
79 \( 1 - 2.69iT - 79T^{2} \)
83 \( 1 + (-5.03 - 5.03i)T + 83iT^{2} \)
89 \( 1 - 12.0T + 89T^{2} \)
97 \( 1 + (-2.23 + 2.23i)T - 97iT^{2} \)
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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

−10.66018019947938541655244321818, −9.832871204909464879597223386813, −9.159035933120682468278011982372, −7.895992450748652744070939660736, −7.40096070541897835208392135194, −5.56510006337225246709950688745, −4.96888831723833553680745392406, −3.58580451676533804929722484719, −2.09640306401550932988616102507, −0.19886582946971344879907106738, 2.19692393287418230212430902959, 3.51482218232164946808939034486, 5.36269548089650130462919142244, 6.14987451860690931474702934838, 6.95669465056238439673140029466, 7.75268580820387789888393165810, 8.985590796923454470249355812712, 9.769685453794973720525400170443, 10.63290149499921760006264034045, 11.39944947744436999127677512345

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