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Properties

Label 2-430-215.49-c1-0-12
Degree $2$
Conductor $430$
Sign $0.987 - 0.155i$
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 + (2.36 + 1.36i)3-s − 4-s + (2.17 + 0.537i)5-s + (1.36 − 2.36i)6-s + (−2.77 + 1.60i)7-s + i·8-s + (2.21 + 3.83i)9-s + (0.537 − 2.17i)10-s − 0.912·11-s + (−2.36 − 1.36i)12-s + (4.63 − 2.67i)13-s + (1.60 + 2.77i)14-s + (4.39 + 4.22i)15-s + 16-s + (1.04 − 0.604i)17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (1.36 + 0.786i)3-s − 0.5·4-s + (0.970 + 0.240i)5-s + (0.556 − 0.963i)6-s + (−1.04 + 0.606i)7-s + 0.353i·8-s + (0.738 + 1.27i)9-s + (0.169 − 0.686i)10-s − 0.275·11-s + (−0.681 − 0.393i)12-s + (1.28 − 0.742i)13-s + (0.428 + 0.742i)14-s + (1.13 + 1.09i)15-s + 0.250·16-s + (0.253 − 0.146i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 - 0.155i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.987 - 0.155i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(430\)    =    \(2 \cdot 5 \cdot 43\)
Sign: $0.987 - 0.155i$
Analytic conductor: \(3.43356\)
Root analytic conductor: \(1.85298\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{430} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 430,\ (\ :1/2),\ 0.987 - 0.155i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.15260 + 0.168120i\)
\(L(\frac12)\) \(\approx\) \(2.15260 + 0.168120i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.17 - 0.537i)T \)
43 \( 1 + (0.613 + 6.52i)T \)
good3 \( 1 + (-2.36 - 1.36i)T + (1.5 + 2.59i)T^{2} \)
7 \( 1 + (2.77 - 1.60i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + 0.912T + 11T^{2} \)
13 \( 1 + (-4.63 + 2.67i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1.04 + 0.604i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.17 - 2.04i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.0863 + 0.0498i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.05 + 7.02i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (4.12 - 7.14i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.0240 + 0.0139i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 0.174T + 41T^{2} \)
47 \( 1 + 7.23iT - 47T^{2} \)
53 \( 1 + (-1.00 - 0.582i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 - 1.44T + 59T^{2} \)
61 \( 1 + (3.43 + 5.95i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (14.0 + 8.09i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (0.176 + 0.306i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-10.0 + 5.77i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (7.17 + 12.4i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-9.42 - 5.44i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (2.18 - 3.78i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 17.1iT - 97T^{2} \)
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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

−10.67336626772290880312089625464, −10.24989678632307386912326316119, −9.332224150654804292863371447800, −8.927161063688924610027388157290, −7.928545879975992779003835150733, −6.29224526715608071177875932201, −5.32866837452909263511883599662, −3.69978391233959430780309041522, −3.10753057982716449858673417019, −2.06052713283594362222649557710, 1.48624729291320279659259017351, 2.96694339552429525386792890186, 4.07391629183749039378944185865, 5.79125992895673267580862034034, 6.64038264448595145993936323192, 7.35510811125121330239917617707, 8.459963673589397238610113998112, 9.142617307290678735702265436298, 9.747651183367636924258770992413, 10.95050353983595811684543447311

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