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Properties

Label 2-430-43.31-c1-0-5
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: $\chi_{430} (31, \cdot )$
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

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

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