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Properties

Label 2-430-43.21-c1-0-11
Degree $2$
Conductor $430$
Sign $0.958 + 0.284i$
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 + (2.65 − 1.28i)3-s + (0.623 + 0.781i)4-s + (−0.222 − 0.974i)5-s + 2.95·6-s − 1.54·7-s + (0.222 + 0.974i)8-s + (3.56 − 4.46i)9-s + (0.222 − 0.974i)10-s + (−1.13 + 1.42i)11-s + (2.65 + 1.28i)12-s + (0.551 + 2.41i)13-s + (−1.38 − 0.669i)14-s + (−1.84 − 2.30i)15-s + (−0.222 + 0.974i)16-s + (1.45 − 6.36i)17-s + ⋯
L(s)  = 1  + (0.637 + 0.306i)2-s + (1.53 − 0.739i)3-s + (0.311 + 0.390i)4-s + (−0.0995 − 0.436i)5-s + 1.20·6-s − 0.583·7-s + (0.0786 + 0.344i)8-s + (1.18 − 1.48i)9-s + (0.0703 − 0.308i)10-s + (−0.342 + 0.429i)11-s + (0.767 + 0.369i)12-s + (0.153 + 0.670i)13-s + (−0.371 − 0.178i)14-s + (−0.475 − 0.595i)15-s + (−0.0556 + 0.243i)16-s + (0.352 − 1.54i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.958 + 0.284i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.958 + 0.284i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(430\)    =    \(2 \cdot 5 \cdot 43\)
Sign: $0.958 + 0.284i$
Analytic conductor: \(3.43356\)
Root analytic conductor: \(1.85298\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{430} (21, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 430,\ (\ :1/2),\ 0.958 + 0.284i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.88173 - 0.418180i\)
\(L(\frac12)\) \(\approx\) \(2.88173 - 0.418180i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.222 + 0.974i)T \)
43 \( 1 + (6.06 - 2.49i)T \)
good3 \( 1 + (-2.65 + 1.28i)T + (1.87 - 2.34i)T^{2} \)
7 \( 1 + 1.54T + 7T^{2} \)
11 \( 1 + (1.13 - 1.42i)T + (-2.44 - 10.7i)T^{2} \)
13 \( 1 + (-0.551 - 2.41i)T + (-11.7 + 5.64i)T^{2} \)
17 \( 1 + (-1.45 + 6.36i)T + (-15.3 - 7.37i)T^{2} \)
19 \( 1 + (-2.73 - 3.42i)T + (-4.22 + 18.5i)T^{2} \)
23 \( 1 + (3.74 - 4.69i)T + (-5.11 - 22.4i)T^{2} \)
29 \( 1 + (5.05 + 2.43i)T + (18.0 + 22.6i)T^{2} \)
31 \( 1 + (1.25 + 0.603i)T + (19.3 + 24.2i)T^{2} \)
37 \( 1 + 3.65T + 37T^{2} \)
41 \( 1 + (-3.63 - 1.75i)T + (25.5 + 32.0i)T^{2} \)
47 \( 1 + (-0.117 - 0.146i)T + (-10.4 + 45.8i)T^{2} \)
53 \( 1 + (1.91 - 8.40i)T + (-47.7 - 22.9i)T^{2} \)
59 \( 1 + (-1.78 + 7.81i)T + (-53.1 - 25.5i)T^{2} \)
61 \( 1 + (-4.46 + 2.14i)T + (38.0 - 47.6i)T^{2} \)
67 \( 1 + (1.04 + 1.31i)T + (-14.9 + 65.3i)T^{2} \)
71 \( 1 + (-5.86 - 7.34i)T + (-15.7 + 69.2i)T^{2} \)
73 \( 1 + (-1.41 - 6.19i)T + (-65.7 + 31.6i)T^{2} \)
79 \( 1 - 14.3T + 79T^{2} \)
83 \( 1 + (-14.4 + 6.93i)T + (51.7 - 64.8i)T^{2} \)
89 \( 1 + (1.93 - 0.932i)T + (55.4 - 69.5i)T^{2} \)
97 \( 1 + (0.385 - 0.483i)T + (-21.5 - 94.5i)T^{2} \)
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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

−11.53369584449450535912098828710, −9.682365698801416966320642359281, −9.348906414361647231193324881216, −8.064996574114359512927618988838, −7.54145085735282264248737233107, −6.67515083488363578424434491769, −5.34788139417078497865480280186, −3.94146925556337002249883802757, −3.05988243059719360370999676047, −1.82252743008900809054394463957, 2.24826218001248133290038849444, 3.33972363135397021444647134625, 3.79053051710380163241405334676, 5.20439426623732991471070538151, 6.45215276359277535846779848914, 7.73363572651340453331926916465, 8.509353184602775703811309973434, 9.512741855946489534237485201085, 10.35726000650726461841592343098, 10.86049390400400204488498088547

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