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Properties

Label 2-430-43.4-c1-0-8
Degree $2$
Conductor $430$
Sign $0.676 + 0.736i$
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 + (0.457 + 0.573i)3-s + (−0.222 − 0.974i)4-s + (0.900 + 0.433i)5-s + 0.733·6-s − 0.660·7-s + (−0.900 − 0.433i)8-s + (0.547 − 2.40i)9-s + (0.900 − 0.433i)10-s + (0.533 − 2.33i)11-s + (0.457 − 0.573i)12-s + (3.57 + 1.72i)13-s + (−0.411 + 0.516i)14-s + (0.163 + 0.714i)15-s + (−0.900 + 0.433i)16-s + (5.85 − 2.81i)17-s + ⋯
L(s)  = 1  + (0.440 − 0.552i)2-s + (0.263 + 0.330i)3-s + (−0.111 − 0.487i)4-s + (0.402 + 0.194i)5-s + 0.299·6-s − 0.249·7-s + (−0.318 − 0.153i)8-s + (0.182 − 0.800i)9-s + (0.284 − 0.137i)10-s + (0.160 − 0.704i)11-s + (0.131 − 0.165i)12-s + (0.991 + 0.477i)13-s + (−0.110 + 0.138i)14-s + (0.0421 + 0.184i)15-s + (−0.225 + 0.108i)16-s + (1.41 − 0.683i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.676 + 0.736i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.676 + 0.736i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(430\)    =    \(2 \cdot 5 \cdot 43\)
Sign: $0.676 + 0.736i$
Analytic conductor: \(3.43356\)
Root analytic conductor: \(1.85298\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{430} (391, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 430,\ (\ :1/2),\ 0.676 + 0.736i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.85372 - 0.814683i\)
\(L(\frac12)\) \(\approx\) \(1.85372 - 0.814683i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.900 - 0.433i)T \)
43 \( 1 + (5.65 + 3.31i)T \)
good3 \( 1 + (-0.457 - 0.573i)T + (-0.667 + 2.92i)T^{2} \)
7 \( 1 + 0.660T + 7T^{2} \)
11 \( 1 + (-0.533 + 2.33i)T + (-9.91 - 4.77i)T^{2} \)
13 \( 1 + (-3.57 - 1.72i)T + (8.10 + 10.1i)T^{2} \)
17 \( 1 + (-5.85 + 2.81i)T + (10.5 - 13.2i)T^{2} \)
19 \( 1 + (0.0968 + 0.424i)T + (-17.1 + 8.24i)T^{2} \)
23 \( 1 + (0.856 - 3.75i)T + (-20.7 - 9.97i)T^{2} \)
29 \( 1 + (4.50 - 5.64i)T + (-6.45 - 28.2i)T^{2} \)
31 \( 1 + (2.99 - 3.75i)T + (-6.89 - 30.2i)T^{2} \)
37 \( 1 + 1.29T + 37T^{2} \)
41 \( 1 + (-2.21 + 2.77i)T + (-9.12 - 39.9i)T^{2} \)
47 \( 1 + (1.34 + 5.89i)T + (-42.3 + 20.3i)T^{2} \)
53 \( 1 + (4.89 - 2.35i)T + (33.0 - 41.4i)T^{2} \)
59 \( 1 + (8.41 - 4.05i)T + (36.7 - 46.1i)T^{2} \)
61 \( 1 + (-4.20 - 5.26i)T + (-13.5 + 59.4i)T^{2} \)
67 \( 1 + (-0.134 - 0.589i)T + (-60.3 + 29.0i)T^{2} \)
71 \( 1 + (-2.10 - 9.20i)T + (-63.9 + 30.8i)T^{2} \)
73 \( 1 + (1.89 + 0.911i)T + (45.5 + 57.0i)T^{2} \)
79 \( 1 + 0.302T + 79T^{2} \)
83 \( 1 + (3.10 + 3.89i)T + (-18.4 + 80.9i)T^{2} \)
89 \( 1 + (-7.38 - 9.26i)T + (-19.8 + 86.7i)T^{2} \)
97 \( 1 + (1.13 - 4.96i)T + (-87.3 - 42.0i)T^{2} \)
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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

−11.09293723428870959208540134009, −10.14112079181177214972840861552, −9.372627956238837990382363811789, −8.675261826682028987165840134427, −7.16082563902324999796166410960, −6.13139689538529581010456742784, −5.25870098701560361012613880452, −3.70085378270601783392351304350, −3.22072875039742789704655608311, −1.37278414176334741098985867710, 1.78720467174923202469342431451, 3.31556133793258900558745361785, 4.56381686017928694986416649805, 5.69153720611195841578319127582, 6.46620540859782551367688655054, 7.75228761786466563553130828461, 8.152760701523034033834014082516, 9.440846305419450982550368234817, 10.28489137000072959621277568399, 11.34458437308885419570020208720

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