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Properties

Label 2-430-43.21-c1-0-5
Degree $2$
Conductor $430$
Sign $0.835 + 0.549i$
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 + (1.86 − 0.896i)3-s + (0.623 + 0.781i)4-s + (0.222 + 0.974i)5-s − 2.06·6-s + 1.71·7-s + (−0.222 − 0.974i)8-s + (0.789 − 0.989i)9-s + (0.222 − 0.974i)10-s + (1.15 − 1.44i)11-s + (1.86 + 0.896i)12-s + (−0.0681 − 0.298i)13-s + (−1.54 − 0.742i)14-s + (1.28 + 1.61i)15-s + (−0.222 + 0.974i)16-s + (0.777 − 3.40i)17-s + ⋯
L(s)  = 1  + (−0.637 − 0.306i)2-s + (1.07 − 0.517i)3-s + (0.311 + 0.390i)4-s + (0.0995 + 0.436i)5-s − 0.843·6-s + 0.646·7-s + (−0.0786 − 0.344i)8-s + (0.263 − 0.329i)9-s + (0.0703 − 0.308i)10-s + (0.346 − 0.434i)11-s + (0.537 + 0.258i)12-s + (−0.0188 − 0.0827i)13-s + (−0.411 − 0.198i)14-s + (0.332 + 0.416i)15-s + (−0.0556 + 0.243i)16-s + (0.188 − 0.826i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.835 + 0.549i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.835 + 0.549i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(430\)    =    \(2 \cdot 5 \cdot 43\)
Sign: $0.835 + 0.549i$
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: \((2,\ 430,\ (\ :1/2),\ 0.835 + 0.549i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.52958 - 0.457489i\)
\(L(\frac12)\) \(\approx\) \(1.52958 - 0.457489i\)
\(L(\frac{3}{2})\) 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 \( 1 + (6.52 - 0.642i)T \)
good3 \( 1 + (-1.86 + 0.896i)T + (1.87 - 2.34i)T^{2} \)
7 \( 1 - 1.71T + 7T^{2} \)
11 \( 1 + (-1.15 + 1.44i)T + (-2.44 - 10.7i)T^{2} \)
13 \( 1 + (0.0681 + 0.298i)T + (-11.7 + 5.64i)T^{2} \)
17 \( 1 + (-0.777 + 3.40i)T + (-15.3 - 7.37i)T^{2} \)
19 \( 1 + (-4.50 - 5.65i)T + (-4.22 + 18.5i)T^{2} \)
23 \( 1 + (-2.14 + 2.68i)T + (-5.11 - 22.4i)T^{2} \)
29 \( 1 + (-1.94 - 0.935i)T + (18.0 + 22.6i)T^{2} \)
31 \( 1 + (7.13 + 3.43i)T + (19.3 + 24.2i)T^{2} \)
37 \( 1 - 9.07T + 37T^{2} \)
41 \( 1 + (8.44 + 4.06i)T + (25.5 + 32.0i)T^{2} \)
47 \( 1 + (-2.29 - 2.87i)T + (-10.4 + 45.8i)T^{2} \)
53 \( 1 + (-1.31 + 5.74i)T + (-47.7 - 22.9i)T^{2} \)
59 \( 1 + (-0.744 + 3.26i)T + (-53.1 - 25.5i)T^{2} \)
61 \( 1 + (0.422 - 0.203i)T + (38.0 - 47.6i)T^{2} \)
67 \( 1 + (-2.52 - 3.16i)T + (-14.9 + 65.3i)T^{2} \)
71 \( 1 + (3.40 + 4.27i)T + (-15.7 + 69.2i)T^{2} \)
73 \( 1 + (-1.35 - 5.91i)T + (-65.7 + 31.6i)T^{2} \)
79 \( 1 + 2.70T + 79T^{2} \)
83 \( 1 + (5.99 - 2.88i)T + (51.7 - 64.8i)T^{2} \)
89 \( 1 + (8.64 - 4.16i)T + (55.4 - 69.5i)T^{2} \)
97 \( 1 + (3.83 - 4.80i)T + (-21.5 - 94.5i)T^{2} \)
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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

−11.11307547080466858237234469947, −10.00319611229010499874189081696, −9.197445261162399018624849376405, −8.258680335414439536590486402370, −7.70639130406131263677320285639, −6.80096577657298008415026249676, −5.38402782436376983238549355301, −3.64768889363589758258571939207, −2.68746187480224821006183108910, −1.48109700929931956811039256553, 1.56553700649908362890993698784, 3.04741520183178083561268903664, 4.38526063505027359958772027637, 5.43955595083772691566414961728, 6.86757934709214417771032048603, 7.85997257987283618395702033508, 8.625312654309852124016871519952, 9.308329042357305918024765176702, 9.941542378026541034135372688646, 11.10209935989765126082412197937

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