Properties

Label 2-430-43.25-c1-0-10
Degree $2$
Conductor $430$
Sign $0.808 + 0.588i$
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 + (−0.889 − 0.606i)3-s + (0.623 + 0.781i)4-s + (−0.733 + 0.680i)5-s + (−0.538 − 0.932i)6-s + (2.53 − 4.38i)7-s + (0.222 + 0.974i)8-s + (−0.672 − 1.71i)9-s + (−0.955 + 0.294i)10-s + (−0.488 + 0.612i)11-s + (−0.0804 − 1.07i)12-s + (1.07 + 0.331i)13-s + (4.18 − 2.85i)14-s + (1.06 − 0.160i)15-s + (−0.222 + 0.974i)16-s + (1.63 + 1.51i)17-s + ⋯
L(s)  = 1  + (0.637 + 0.306i)2-s + (−0.513 − 0.350i)3-s + (0.311 + 0.390i)4-s + (−0.327 + 0.304i)5-s + (−0.219 − 0.380i)6-s + (0.956 − 1.65i)7-s + (0.0786 + 0.344i)8-s + (−0.224 − 0.571i)9-s + (−0.302 + 0.0932i)10-s + (−0.147 + 0.184i)11-s + (−0.0232 − 0.309i)12-s + (0.298 + 0.0919i)13-s + (1.11 − 0.761i)14-s + (0.274 − 0.0414i)15-s + (−0.0556 + 0.243i)16-s + (0.396 + 0.368i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.808 + 0.588i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.808 + 0.588i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(430\)    =    \(2 \cdot 5 \cdot 43\)
Sign: $0.808 + 0.588i$
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: \((2,\ 430,\ (\ :1/2),\ 0.808 + 0.588i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.62679 - 0.529163i\)
\(L(\frac12)\) \(\approx\) \(1.62679 - 0.529163i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.12 - 6.46i)T \)
good3 \( 1 + (0.889 + 0.606i)T + (1.09 + 2.79i)T^{2} \)
7 \( 1 + (-2.53 + 4.38i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.488 - 0.612i)T + (-2.44 - 10.7i)T^{2} \)
13 \( 1 + (-1.07 - 0.331i)T + (10.7 + 7.32i)T^{2} \)
17 \( 1 + (-1.63 - 1.51i)T + (1.27 + 16.9i)T^{2} \)
19 \( 1 + (-2.93 + 7.47i)T + (-13.9 - 12.9i)T^{2} \)
23 \( 1 + (-4.37 - 0.658i)T + (21.9 + 6.77i)T^{2} \)
29 \( 1 + (1.11 - 0.758i)T + (10.5 - 26.9i)T^{2} \)
31 \( 1 + (0.00220 + 0.0293i)T + (-30.6 + 4.62i)T^{2} \)
37 \( 1 + (1.20 + 2.08i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-3.88 - 1.87i)T + (25.5 + 32.0i)T^{2} \)
47 \( 1 + (1.27 + 1.59i)T + (-10.4 + 45.8i)T^{2} \)
53 \( 1 + (9.64 - 2.97i)T + (43.7 - 29.8i)T^{2} \)
59 \( 1 + (0.663 - 2.90i)T + (-53.1 - 25.5i)T^{2} \)
61 \( 1 + (0.225 - 3.00i)T + (-60.3 - 9.09i)T^{2} \)
67 \( 1 + (3.94 - 10.0i)T + (-49.1 - 45.5i)T^{2} \)
71 \( 1 + (-0.0568 + 0.00857i)T + (67.8 - 20.9i)T^{2} \)
73 \( 1 + (5.82 + 1.79i)T + (60.3 + 41.1i)T^{2} \)
79 \( 1 + (-3.48 + 6.02i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-8.12 - 5.53i)T + (30.3 + 77.2i)T^{2} \)
89 \( 1 + (-8.44 - 5.76i)T + (32.5 + 82.8i)T^{2} \)
97 \( 1 + (8.25 - 10.3i)T + (-21.5 - 94.5i)T^{2} \)
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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

−11.18660539366866670911400206228, −10.63309316485384694804048809986, −9.207090976184539247550653856266, −7.87334240596921282195588089249, −7.21036876088599384035666417085, −6.53813638394160743024451402516, −5.20275670468095814603344590693, −4.31139043660290339083963800644, −3.20901206292608182459576519891, −1.07275001000606830579616044205, 1.82712431218553248367487232418, 3.20016686871499277696081197672, 4.70947810374950357226101204790, 5.38996587052253014171117032101, 5.97764282058727842338398037482, 7.73847372030895777278813062579, 8.464149569896934761793069460451, 9.541918322497950700670395420100, 10.70208256653017590475255851654, 11.42064138399461348145509036385

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