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Properties

Label 2-430-43.25-c1-0-0
Degree $2$
Conductor $430$
Sign $-0.711 - 0.702i$
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 + (0.492 + 0.335i)3-s + (0.623 + 0.781i)4-s + (−0.733 + 0.680i)5-s + (−0.298 − 0.516i)6-s + (−0.982 + 1.70i)7-s + (−0.222 − 0.974i)8-s + (−0.966 − 2.46i)9-s + (0.955 − 0.294i)10-s + (−3.99 + 5.00i)11-s + (0.0445 + 0.594i)12-s + (0.766 + 0.236i)13-s + (1.62 − 1.10i)14-s + (−0.589 + 0.0888i)15-s + (−0.222 + 0.974i)16-s + (−3.05 − 2.83i)17-s + ⋯
L(s)  = 1  + (−0.637 − 0.306i)2-s + (0.284 + 0.193i)3-s + (0.311 + 0.390i)4-s + (−0.327 + 0.304i)5-s + (−0.121 − 0.210i)6-s + (−0.371 + 0.643i)7-s + (−0.0786 − 0.344i)8-s + (−0.322 − 0.820i)9-s + (0.302 − 0.0932i)10-s + (−1.20 + 1.51i)11-s + (0.0128 + 0.171i)12-s + (0.212 + 0.0655i)13-s + (0.434 − 0.295i)14-s + (−0.152 + 0.0229i)15-s + (−0.0556 + 0.243i)16-s + (−0.740 − 0.686i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.711 - 0.702i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.711 - 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(430\)    =    \(2 \cdot 5 \cdot 43\)
Sign: $-0.711 - 0.702i$
Analytic conductor: \(3.43356\)
Root analytic conductor: \(1.85298\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{430} (111, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 430,\ (\ :1/2),\ -0.711 - 0.702i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.165139 + 0.402522i\)
\(L(\frac12)\) \(\approx\) \(0.165139 + 0.402522i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.733 - 0.680i)T \)
43 \( 1 + (2.07 - 6.22i)T \)
good3 \( 1 + (-0.492 - 0.335i)T + (1.09 + 2.79i)T^{2} \)
7 \( 1 + (0.982 - 1.70i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (3.99 - 5.00i)T + (-2.44 - 10.7i)T^{2} \)
13 \( 1 + (-0.766 - 0.236i)T + (10.7 + 7.32i)T^{2} \)
17 \( 1 + (3.05 + 2.83i)T + (1.27 + 16.9i)T^{2} \)
19 \( 1 + (0.838 - 2.13i)T + (-13.9 - 12.9i)T^{2} \)
23 \( 1 + (2.14 + 0.322i)T + (21.9 + 6.77i)T^{2} \)
29 \( 1 + (5.99 - 4.08i)T + (10.5 - 26.9i)T^{2} \)
31 \( 1 + (-0.318 - 4.24i)T + (-30.6 + 4.62i)T^{2} \)
37 \( 1 + (4.21 + 7.30i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-10.2 - 4.93i)T + (25.5 + 32.0i)T^{2} \)
47 \( 1 + (-2.27 - 2.85i)T + (-10.4 + 45.8i)T^{2} \)
53 \( 1 + (9.96 - 3.07i)T + (43.7 - 29.8i)T^{2} \)
59 \( 1 + (0.680 - 2.98i)T + (-53.1 - 25.5i)T^{2} \)
61 \( 1 + (0.426 - 5.68i)T + (-60.3 - 9.09i)T^{2} \)
67 \( 1 + (-3.24 + 8.27i)T + (-49.1 - 45.5i)T^{2} \)
71 \( 1 + (-2.80 + 0.422i)T + (67.8 - 20.9i)T^{2} \)
73 \( 1 + (-10.6 - 3.28i)T + (60.3 + 41.1i)T^{2} \)
79 \( 1 + (-6.22 + 10.7i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (8.89 + 6.06i)T + (30.3 + 77.2i)T^{2} \)
89 \( 1 + (4.66 + 3.18i)T + (32.5 + 82.8i)T^{2} \)
97 \( 1 + (-1.24 + 1.55i)T + (-21.5 - 94.5i)T^{2} \)
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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

−11.35232726992633788463644658608, −10.51856824584737591981653164057, −9.543756850626179921537632801626, −9.048196372824396493933759746330, −7.895572053368652616016839400293, −7.10116787637062550444986284778, −5.97262899376154736084449108654, −4.53939745766705667253343247720, −3.20900297317971970926593279269, −2.21267731022553834545313916530, 0.30300010574321133573539929819, 2.31813162736871691252876237443, 3.71666051077243192367752689517, 5.22388231624540187505147032433, 6.17486778391067268471735343809, 7.40203793640327146521611917296, 8.182555181920969289503648114615, 8.663990146219054323366047188097, 9.916528689305160493726020812813, 10.94565310286288637199143865606

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