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Properties

Label 2-430-43.11-c1-0-7
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: $\chi_{430} (11, \cdot )$
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

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

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