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Properties

Label 2-430-43.41-c1-0-1
Degree $2$
Conductor $430$
Sign $0.999 - 0.0187i$
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 + (−0.991 − 0.477i)3-s + (0.623 − 0.781i)4-s + (−0.222 + 0.974i)5-s + 1.10·6-s + 1.63·7-s + (−0.222 + 0.974i)8-s + (−1.11 − 1.39i)9-s + (−0.222 − 0.974i)10-s + (0.880 + 1.10i)11-s + (−0.991 + 0.477i)12-s + (0.400 − 1.75i)13-s + (−1.47 + 0.709i)14-s + (0.686 − 0.860i)15-s + (−0.222 − 0.974i)16-s + (1.00 + 4.40i)17-s + ⋯
L(s)  = 1  + (−0.637 + 0.306i)2-s + (−0.572 − 0.275i)3-s + (0.311 − 0.390i)4-s + (−0.0995 + 0.436i)5-s + 0.449·6-s + 0.618·7-s + (−0.0786 + 0.344i)8-s + (−0.371 − 0.466i)9-s + (−0.0703 − 0.308i)10-s + (0.265 + 0.333i)11-s + (−0.286 + 0.137i)12-s + (0.111 − 0.487i)13-s + (−0.393 + 0.189i)14-s + (0.177 − 0.222i)15-s + (−0.0556 − 0.243i)16-s + (0.243 + 1.06i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0187i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0187i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(430\)    =    \(2 \cdot 5 \cdot 43\)
Sign: $0.999 - 0.0187i$
Analytic conductor: \(3.43356\)
Root analytic conductor: \(1.85298\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{430} (41, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 430,\ (\ :1/2),\ 0.999 - 0.0187i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.888128 + 0.00832868i\)
\(L(\frac12)\) \(\approx\) \(0.888128 + 0.00832868i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-5.17 - 4.02i)T \)
good3 \( 1 + (0.991 + 0.477i)T + (1.87 + 2.34i)T^{2} \)
7 \( 1 - 1.63T + 7T^{2} \)
11 \( 1 + (-0.880 - 1.10i)T + (-2.44 + 10.7i)T^{2} \)
13 \( 1 + (-0.400 + 1.75i)T + (-11.7 - 5.64i)T^{2} \)
17 \( 1 + (-1.00 - 4.40i)T + (-15.3 + 7.37i)T^{2} \)
19 \( 1 + (-2.89 + 3.63i)T + (-4.22 - 18.5i)T^{2} \)
23 \( 1 + (-0.414 - 0.519i)T + (-5.11 + 22.4i)T^{2} \)
29 \( 1 + (-9.23 + 4.44i)T + (18.0 - 22.6i)T^{2} \)
31 \( 1 + (-5.89 + 2.84i)T + (19.3 - 24.2i)T^{2} \)
37 \( 1 - 2.34T + 37T^{2} \)
41 \( 1 + (-10.9 + 5.27i)T + (25.5 - 32.0i)T^{2} \)
47 \( 1 + (3.02 - 3.79i)T + (-10.4 - 45.8i)T^{2} \)
53 \( 1 + (0.561 + 2.46i)T + (-47.7 + 22.9i)T^{2} \)
59 \( 1 + (-1.55 - 6.79i)T + (-53.1 + 25.5i)T^{2} \)
61 \( 1 + (3.66 + 1.76i)T + (38.0 + 47.6i)T^{2} \)
67 \( 1 + (5.45 - 6.83i)T + (-14.9 - 65.3i)T^{2} \)
71 \( 1 + (4.02 - 5.04i)T + (-15.7 - 69.2i)T^{2} \)
73 \( 1 + (-3.55 + 15.5i)T + (-65.7 - 31.6i)T^{2} \)
79 \( 1 - 3.26T + 79T^{2} \)
83 \( 1 + (-0.418 - 0.201i)T + (51.7 + 64.8i)T^{2} \)
89 \( 1 + (15.8 + 7.62i)T + (55.4 + 69.5i)T^{2} \)
97 \( 1 + (1.02 + 1.28i)T + (-21.5 + 94.5i)T^{2} \)
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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

−11.15747361844240215483692942456, −10.32044180796459761165040476482, −9.344221350731850725091459753108, −8.299650936688300278515717848997, −7.52254708709364554811661806642, −6.45157370131921426527123921740, −5.80419865690498354905938360468, −4.46525570383356602699127428885, −2.79583037649006491437786795084, −1.01430385369430847392285961720, 1.12421498326222749629108926643, 2.82712898067304136270793702278, 4.42492899148709886431770380261, 5.30555945157730132873507846174, 6.48410768003279753177675228244, 7.75350124153687726442394480924, 8.449819270781363617142912526885, 9.410283368708439184550663229190, 10.32540066545268006145001765208, 11.21030513855357850765182620890

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