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Properties

Label 2-430-43.21-c1-0-1
Degree $2$
Conductor $430$
Sign $-0.745 - 0.666i$
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  + (0.900 + 0.433i)2-s + (−0.400 + 0.192i)3-s + (0.623 + 0.781i)4-s + (0.222 + 0.974i)5-s − 0.444·6-s − 4.95·7-s + (0.222 + 0.974i)8-s + (−1.74 + 2.19i)9-s + (−0.222 + 0.974i)10-s + (−1.11 + 1.39i)11-s + (−0.400 − 0.192i)12-s + (0.610 + 2.67i)13-s + (−4.46 − 2.15i)14-s + (−0.276 − 0.347i)15-s + (−0.222 + 0.974i)16-s + (0.507 − 2.22i)17-s + ⋯
L(s)  = 1  + (0.637 + 0.306i)2-s + (−0.231 + 0.111i)3-s + (0.311 + 0.390i)4-s + (0.0995 + 0.436i)5-s − 0.181·6-s − 1.87·7-s + (0.0786 + 0.344i)8-s + (−0.582 + 0.730i)9-s + (−0.0703 + 0.308i)10-s + (−0.335 + 0.420i)11-s + (−0.115 − 0.0556i)12-s + (0.169 + 0.741i)13-s + (−1.19 − 0.575i)14-s + (−0.0715 − 0.0896i)15-s + (−0.0556 + 0.243i)16-s + (0.123 − 0.539i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.745 - 0.666i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.745 - 0.666i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(430\)    =    \(2 \cdot 5 \cdot 43\)
Sign: $-0.745 - 0.666i$
Analytic conductor: \(3.43356\)
Root analytic conductor: \(1.85298\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{430} (21, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 430,\ (\ :1/2),\ -0.745 - 0.666i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.409799 + 1.07346i\)
\(L(\frac12)\) \(\approx\) \(0.409799 + 1.07346i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.900 - 0.433i)T \)
5 \( 1 + (-0.222 - 0.974i)T \)
43 \( 1 + (6.54 + 0.327i)T \)
good3 \( 1 + (0.400 - 0.192i)T + (1.87 - 2.34i)T^{2} \)
7 \( 1 + 4.95T + 7T^{2} \)
11 \( 1 + (1.11 - 1.39i)T + (-2.44 - 10.7i)T^{2} \)
13 \( 1 + (-0.610 - 2.67i)T + (-11.7 + 5.64i)T^{2} \)
17 \( 1 + (-0.507 + 2.22i)T + (-15.3 - 7.37i)T^{2} \)
19 \( 1 + (-2.29 - 2.87i)T + (-4.22 + 18.5i)T^{2} \)
23 \( 1 + (-0.765 + 0.959i)T + (-5.11 - 22.4i)T^{2} \)
29 \( 1 + (-7.17 - 3.45i)T + (18.0 + 22.6i)T^{2} \)
31 \( 1 + (0.638 + 0.307i)T + (19.3 + 24.2i)T^{2} \)
37 \( 1 - 5.24T + 37T^{2} \)
41 \( 1 + (2.38 + 1.14i)T + (25.5 + 32.0i)T^{2} \)
47 \( 1 + (4.49 + 5.63i)T + (-10.4 + 45.8i)T^{2} \)
53 \( 1 + (0.614 - 2.69i)T + (-47.7 - 22.9i)T^{2} \)
59 \( 1 + (3.33 - 14.6i)T + (-53.1 - 25.5i)T^{2} \)
61 \( 1 + (5.39 - 2.59i)T + (38.0 - 47.6i)T^{2} \)
67 \( 1 + (-4.55 - 5.70i)T + (-14.9 + 65.3i)T^{2} \)
71 \( 1 + (3.80 + 4.76i)T + (-15.7 + 69.2i)T^{2} \)
73 \( 1 + (-0.660 - 2.89i)T + (-65.7 + 31.6i)T^{2} \)
79 \( 1 - 12.4T + 79T^{2} \)
83 \( 1 + (-15.2 + 7.33i)T + (51.7 - 64.8i)T^{2} \)
89 \( 1 + (2.62 - 1.26i)T + (55.4 - 69.5i)T^{2} \)
97 \( 1 + (-1.28 + 1.61i)T + (-21.5 - 94.5i)T^{2} \)
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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

−11.71075128201632616390557567975, −10.52678915270870148489575128380, −9.887527842679915601538786158198, −8.818833290102204058348874847608, −7.50676661675018521894124879938, −6.64850664530966537383358557633, −5.94495934275392582514436798052, −4.83698812046349392753954364076, −3.46974078475491689390622927704, −2.59199103293075114904932352755, 0.58514697880682382375908681402, 2.91289710018509035127379309488, 3.53070644517605600280741862417, 5.09701123040509808475100099355, 6.14956867799536167103842663962, 6.55430063036039224938590505613, 8.091070066062224990064392659659, 9.282468364194961990993426619108, 9.902832926320845599480778092536, 10.90978611409488817387041476790

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