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Properties

Label 2-430-43.15-c1-0-14
Degree $2$
Conductor $430$
Sign $-0.716 + 0.698i$
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  + (0.623 − 0.781i)2-s + (0.808 − 2.06i)3-s + (−0.222 − 0.974i)4-s + (−0.0747 − 0.997i)5-s + (−1.10 − 1.91i)6-s + (1.63 − 2.83i)7-s + (−0.900 − 0.433i)8-s + (−1.39 − 1.29i)9-s + (−0.826 − 0.563i)10-s + (−0.960 + 4.20i)11-s + (−2.18 − 0.330i)12-s + (1.49 − 1.01i)13-s + (−1.19 − 3.04i)14-s + (−2.11 − 0.652i)15-s + (−0.900 + 0.433i)16-s + (−0.517 + 6.90i)17-s + ⋯
L(s)  = 1  + (0.440 − 0.552i)2-s + (0.467 − 1.18i)3-s + (−0.111 − 0.487i)4-s + (−0.0334 − 0.445i)5-s + (−0.451 − 0.782i)6-s + (0.618 − 1.07i)7-s + (−0.318 − 0.153i)8-s + (−0.464 − 0.431i)9-s + (−0.261 − 0.178i)10-s + (−0.289 + 1.26i)11-s + (−0.632 − 0.0952i)12-s + (0.413 − 0.281i)13-s + (−0.319 − 0.814i)14-s + (−0.546 − 0.168i)15-s + (−0.225 + 0.108i)16-s + (−0.125 + 1.67i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.716 + 0.698i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.716 + 0.698i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(430\)    =    \(2 \cdot 5 \cdot 43\)
Sign: $-0.716 + 0.698i$
Analytic conductor: \(3.43356\)
Root analytic conductor: \(1.85298\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{430} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 430,\ (\ :1/2),\ -0.716 + 0.698i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.771259 - 1.89586i\)
\(L(\frac12)\) \(\approx\) \(0.771259 - 1.89586i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.623 + 0.781i)T \)
5 \( 1 + (0.0747 + 0.997i)T \)
43 \( 1 + (-6.03 + 2.55i)T \)
good3 \( 1 + (-0.808 + 2.06i)T + (-2.19 - 2.04i)T^{2} \)
7 \( 1 + (-1.63 + 2.83i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.960 - 4.20i)T + (-9.91 - 4.77i)T^{2} \)
13 \( 1 + (-1.49 + 1.01i)T + (4.74 - 12.1i)T^{2} \)
17 \( 1 + (0.517 - 6.90i)T + (-16.8 - 2.53i)T^{2} \)
19 \( 1 + (0.235 - 0.218i)T + (1.41 - 18.9i)T^{2} \)
23 \( 1 + (-1.76 + 0.545i)T + (19.0 - 12.9i)T^{2} \)
29 \( 1 + (0.974 + 2.48i)T + (-21.2 + 19.7i)T^{2} \)
31 \( 1 + (1.49 + 0.224i)T + (29.6 + 9.13i)T^{2} \)
37 \( 1 + (2.51 + 4.35i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (3.32 - 4.16i)T + (-9.12 - 39.9i)T^{2} \)
47 \( 1 + (-0.126 - 0.555i)T + (-42.3 + 20.3i)T^{2} \)
53 \( 1 + (-3.12 - 2.12i)T + (19.3 + 49.3i)T^{2} \)
59 \( 1 + (-2.14 + 1.03i)T + (36.7 - 46.1i)T^{2} \)
61 \( 1 + (12.5 - 1.89i)T + (58.2 - 17.9i)T^{2} \)
67 \( 1 + (8.89 - 8.25i)T + (5.00 - 66.8i)T^{2} \)
71 \( 1 + (-15.7 - 4.84i)T + (58.6 + 39.9i)T^{2} \)
73 \( 1 + (-3.22 + 2.20i)T + (26.6 - 67.9i)T^{2} \)
79 \( 1 + (-1.68 + 2.91i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-4.35 + 11.0i)T + (-60.8 - 56.4i)T^{2} \)
89 \( 1 + (-5.34 + 13.6i)T + (-65.2 - 60.5i)T^{2} \)
97 \( 1 + (2.34 - 10.2i)T + (-87.3 - 42.0i)T^{2} \)
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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

−10.80019998435965284654927598034, −10.22184460510736431181611432323, −8.869571473745363784306151720956, −7.88808409750098193921899109514, −7.28621728252808217352664682598, −6.14191073228180319205534959641, −4.74987011343681440996883196373, −3.84541545918386677307289040617, −2.13949952639704056818589657614, −1.26035725845461324508180684848, 2.71838645911956437749333023009, 3.56088014255884584289872032322, 4.86372095235482093284713218080, 5.52594023024738489165918246870, 6.73126439503812328358387244519, 8.000929892290054028684984311727, 8.882104683354875127277714617182, 9.369807625902865146947920733070, 10.73430733165385762092139773356, 11.36175639622373902518236498596

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