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Properties

Label 2-430-43.40-c1-0-6
Degree $2$
Conductor $430$
Sign $-0.0609 - 0.998i$
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 + (−0.112 + 1.50i)3-s + (0.623 + 0.781i)4-s + (0.955 + 0.294i)5-s + (−0.755 + 1.30i)6-s + (0.234 + 0.406i)7-s + (0.222 + 0.974i)8-s + (0.708 + 0.106i)9-s + (0.733 + 0.680i)10-s + (0.399 − 0.501i)11-s + (−1.24 + 0.851i)12-s + (−2.50 + 2.32i)13-s + (0.0350 + 0.468i)14-s + (−0.552 + 1.40i)15-s + (−0.222 + 0.974i)16-s + (−2.60 + 0.802i)17-s + ⋯
L(s)  = 1  + (0.637 + 0.306i)2-s + (−0.0651 + 0.870i)3-s + (0.311 + 0.390i)4-s + (0.427 + 0.131i)5-s + (−0.308 + 0.534i)6-s + (0.0887 + 0.153i)7-s + (0.0786 + 0.344i)8-s + (0.236 + 0.0355i)9-s + (0.231 + 0.215i)10-s + (0.120 − 0.151i)11-s + (−0.360 + 0.245i)12-s + (−0.695 + 0.645i)13-s + (0.00937 + 0.125i)14-s + (−0.142 + 0.363i)15-s + (−0.0556 + 0.243i)16-s + (−0.631 + 0.194i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0609 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0609 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(430\)    =    \(2 \cdot 5 \cdot 43\)
Sign: $-0.0609 - 0.998i$
Analytic conductor: \(3.43356\)
Root analytic conductor: \(1.85298\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{430} (341, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 430,\ (\ :1/2),\ -0.0609 - 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.43702 + 1.52738i\)
\(L(\frac12)\) \(\approx\) \(1.43702 + 1.52738i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.955 - 0.294i)T \)
43 \( 1 + (5.17 + 4.02i)T \)
good3 \( 1 + (0.112 - 1.50i)T + (-2.96 - 0.447i)T^{2} \)
7 \( 1 + (-0.234 - 0.406i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.399 + 0.501i)T + (-2.44 - 10.7i)T^{2} \)
13 \( 1 + (2.50 - 2.32i)T + (0.971 - 12.9i)T^{2} \)
17 \( 1 + (2.60 - 0.802i)T + (14.0 - 9.57i)T^{2} \)
19 \( 1 + (-6.20 + 0.935i)T + (18.1 - 5.60i)T^{2} \)
23 \( 1 + (3.02 + 7.71i)T + (-16.8 + 15.6i)T^{2} \)
29 \( 1 + (-0.0460 - 0.614i)T + (-28.6 + 4.32i)T^{2} \)
31 \( 1 + (8.76 - 5.97i)T + (11.3 - 28.8i)T^{2} \)
37 \( 1 + (-4.97 + 8.61i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-10.4 - 5.04i)T + (25.5 + 32.0i)T^{2} \)
47 \( 1 + (1.65 + 2.07i)T + (-10.4 + 45.8i)T^{2} \)
53 \( 1 + (-3.24 - 3.01i)T + (3.96 + 52.8i)T^{2} \)
59 \( 1 + (-2.75 + 12.0i)T + (-53.1 - 25.5i)T^{2} \)
61 \( 1 + (-2.24 - 1.53i)T + (22.2 + 56.7i)T^{2} \)
67 \( 1 + (1.14 - 0.171i)T + (64.0 - 19.7i)T^{2} \)
71 \( 1 + (-1.01 + 2.57i)T + (-52.0 - 48.2i)T^{2} \)
73 \( 1 + (-6.69 + 6.21i)T + (5.45 - 72.7i)T^{2} \)
79 \( 1 + (-1.69 - 2.93i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.0477 - 0.636i)T + (-82.0 - 12.3i)T^{2} \)
89 \( 1 + (-1.01 + 13.5i)T + (-88.0 - 13.2i)T^{2} \)
97 \( 1 + (8.58 - 10.7i)T + (-21.5 - 94.5i)T^{2} \)
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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

−11.32470864977517037506894114367, −10.54976949137537822970709350389, −9.591919357893092645777456015251, −8.869658638052606904972714657326, −7.48427290886767524027659015401, −6.62990886947729467103208830877, −5.45449054548453753922122795403, −4.65720532196880834341597348344, −3.67891103117261454360090611000, −2.23313992282315058316303860391, 1.27245899914611267066863590511, 2.50842573308656130590987951068, 3.95702057073592119660484411959, 5.24655401095932030471102452985, 6.06300787856799885822823649518, 7.27211836810111774679185515371, 7.74788184055328437039250655044, 9.435410211694272222326952811376, 9.957516013763332098209550784829, 11.24557737850734736712569794060

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