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Properties

Label 2-430-43.10-c1-0-6
Degree $2$
Conductor $430$
Sign $0.989 + 0.146i$
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  + (−0.222 − 0.974i)2-s + (0.664 + 0.616i)3-s + (−0.900 + 0.433i)4-s + (0.988 − 0.149i)5-s + (0.453 − 0.784i)6-s + (1.53 + 2.65i)7-s + (0.623 + 0.781i)8-s + (−0.162 − 2.17i)9-s + (−0.365 − 0.930i)10-s + (0.187 + 0.0903i)11-s + (−0.865 − 0.267i)12-s + (−0.763 + 1.94i)13-s + (2.24 − 2.08i)14-s + (0.748 + 0.510i)15-s + (0.623 − 0.781i)16-s + (6.42 + 0.969i)17-s + ⋯
L(s)  = 1  + (−0.157 − 0.689i)2-s + (0.383 + 0.355i)3-s + (−0.450 + 0.216i)4-s + (0.442 − 0.0666i)5-s + (0.184 − 0.320i)6-s + (0.580 + 1.00i)7-s + (0.220 + 0.276i)8-s + (−0.0542 − 0.724i)9-s + (−0.115 − 0.294i)10-s + (0.0565 + 0.0272i)11-s + (−0.249 − 0.0771i)12-s + (−0.211 + 0.539i)13-s + (0.601 − 0.557i)14-s + (0.193 + 0.131i)15-s + (0.155 − 0.195i)16-s + (1.55 + 0.235i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 + 0.146i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.989 + 0.146i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(430\)    =    \(2 \cdot 5 \cdot 43\)
Sign: $0.989 + 0.146i$
Analytic conductor: \(3.43356\)
Root analytic conductor: \(1.85298\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{430} (311, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 430,\ (\ :1/2),\ 0.989 + 0.146i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.58905 - 0.116825i\)
\(L(\frac12)\) \(\approx\) \(1.58905 - 0.116825i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.222 + 0.974i)T \)
5 \( 1 + (-0.988 + 0.149i)T \)
43 \( 1 + (6.16 - 2.22i)T \)
good3 \( 1 + (-0.664 - 0.616i)T + (0.224 + 2.99i)T^{2} \)
7 \( 1 + (-1.53 - 2.65i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.187 - 0.0903i)T + (6.85 + 8.60i)T^{2} \)
13 \( 1 + (0.763 - 1.94i)T + (-9.52 - 8.84i)T^{2} \)
17 \( 1 + (-6.42 - 0.969i)T + (16.2 + 5.01i)T^{2} \)
19 \( 1 + (0.288 - 3.85i)T + (-18.7 - 2.83i)T^{2} \)
23 \( 1 + (0.599 - 0.408i)T + (8.40 - 21.4i)T^{2} \)
29 \( 1 + (-2.23 + 2.07i)T + (2.16 - 28.9i)T^{2} \)
31 \( 1 + (0.896 + 0.276i)T + (25.6 + 17.4i)T^{2} \)
37 \( 1 + (-4.40 + 7.62i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.899 - 3.94i)T + (-36.9 + 17.7i)T^{2} \)
47 \( 1 + (-0.376 + 0.181i)T + (29.3 - 36.7i)T^{2} \)
53 \( 1 + (4.80 + 12.2i)T + (-38.8 + 36.0i)T^{2} \)
59 \( 1 + (6.03 - 7.57i)T + (-13.1 - 57.5i)T^{2} \)
61 \( 1 + (9.36 - 2.88i)T + (50.4 - 34.3i)T^{2} \)
67 \( 1 + (-0.838 + 11.1i)T + (-66.2 - 9.98i)T^{2} \)
71 \( 1 + (-3.71 - 2.53i)T + (25.9 + 66.0i)T^{2} \)
73 \( 1 + (3.26 - 8.33i)T + (-53.5 - 49.6i)T^{2} \)
79 \( 1 + (2.18 + 3.78i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-8.53 - 7.91i)T + (6.20 + 82.7i)T^{2} \)
89 \( 1 + (3.02 + 2.80i)T + (6.65 + 88.7i)T^{2} \)
97 \( 1 + (8.36 + 4.02i)T + (60.4 + 75.8i)T^{2} \)
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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

−11.19147367578953136098860084511, −9.971475664865888804187786111801, −9.530718745009509117478195006352, −8.623283285739348768604088963336, −7.83878143444630800821782954438, −6.22547674162364019292668948040, −5.29785055720379858729358730774, −4.02730825274755954905244282107, −2.89483627092071162293596455017, −1.61264053308824306052215955689, 1.28785774703535525055073170085, 3.00338976600874250808919341960, 4.60854401139730481143832540889, 5.44261578868517655194577111709, 6.72692745292981314907778968845, 7.65231217836870826150730825861, 8.072574396072001126488704013932, 9.273059885547100984136264151941, 10.28050063869136145490022722369, 10.87152656449202661473799928100

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