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Properties

Label 2-430-43.9-c1-0-0
Degree $2$
Conductor $430$
Sign $-0.994 + 0.100i$
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 + (−0.399 − 0.0602i)3-s + (−0.222 + 0.974i)4-s + (−0.826 − 0.563i)5-s + (−0.201 − 0.349i)6-s + (−2.60 + 4.51i)7-s + (−0.900 + 0.433i)8-s + (−2.71 − 0.836i)9-s + (−0.0747 − 0.997i)10-s + (−0.576 − 2.52i)11-s + (0.147 − 0.376i)12-s + (0.187 − 2.49i)13-s + (−5.15 + 0.776i)14-s + (0.296 + 0.274i)15-s + (−0.900 − 0.433i)16-s + (−5.31 + 3.62i)17-s + ⋯
L(s)  = 1  + (0.440 + 0.552i)2-s + (−0.230 − 0.0347i)3-s + (−0.111 + 0.487i)4-s + (−0.369 − 0.251i)5-s + (−0.0824 − 0.142i)6-s + (−0.985 + 1.70i)7-s + (−0.318 + 0.153i)8-s + (−0.903 − 0.278i)9-s + (−0.0236 − 0.315i)10-s + (−0.173 − 0.762i)11-s + (0.0425 − 0.108i)12-s + (0.0518 − 0.692i)13-s + (−1.37 + 0.207i)14-s + (0.0764 + 0.0709i)15-s + (−0.225 − 0.108i)16-s + (−1.28 + 0.878i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 + 0.100i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.994 + 0.100i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(430\)    =    \(2 \cdot 5 \cdot 43\)
Sign: $-0.994 + 0.100i$
Analytic conductor: \(3.43356\)
Root analytic conductor: \(1.85298\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{430} (181, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 430,\ (\ :1/2),\ -0.994 + 0.100i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0267738 - 0.532622i\)
\(L(\frac12)\) \(\approx\) \(0.0267738 - 0.532622i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.826 + 0.563i)T \)
43 \( 1 + (-6.36 + 1.58i)T \)
good3 \( 1 + (0.399 + 0.0602i)T + (2.86 + 0.884i)T^{2} \)
7 \( 1 + (2.60 - 4.51i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.576 + 2.52i)T + (-9.91 + 4.77i)T^{2} \)
13 \( 1 + (-0.187 + 2.49i)T + (-12.8 - 1.93i)T^{2} \)
17 \( 1 + (5.31 - 3.62i)T + (6.21 - 15.8i)T^{2} \)
19 \( 1 + (-5.53 + 1.70i)T + (15.6 - 10.7i)T^{2} \)
23 \( 1 + (4.20 - 3.89i)T + (1.71 - 22.9i)T^{2} \)
29 \( 1 + (5.38 - 0.811i)T + (27.7 - 8.54i)T^{2} \)
31 \( 1 + (2.97 - 7.58i)T + (-22.7 - 21.0i)T^{2} \)
37 \( 1 + (-1.11 - 1.93i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-4.58 - 5.74i)T + (-9.12 + 39.9i)T^{2} \)
47 \( 1 + (-1.02 + 4.50i)T + (-42.3 - 20.3i)T^{2} \)
53 \( 1 + (-0.411 - 5.49i)T + (-52.4 + 7.89i)T^{2} \)
59 \( 1 + (-3.29 - 1.58i)T + (36.7 + 46.1i)T^{2} \)
61 \( 1 + (2.68 + 6.84i)T + (-44.7 + 41.4i)T^{2} \)
67 \( 1 + (-13.0 + 4.01i)T + (55.3 - 37.7i)T^{2} \)
71 \( 1 + (-0.993 - 0.922i)T + (5.30 + 70.8i)T^{2} \)
73 \( 1 + (1.06 - 14.2i)T + (-72.1 - 10.8i)T^{2} \)
79 \( 1 + (1.83 - 3.17i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (14.9 + 2.25i)T + (79.3 + 24.4i)T^{2} \)
89 \( 1 + (-0.686 - 0.103i)T + (85.0 + 26.2i)T^{2} \)
97 \( 1 + (-1.94 - 8.51i)T + (-87.3 + 42.0i)T^{2} \)
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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

−11.74920041728072106622807399510, −11.06077329904100494442982680482, −9.468911307721227723249335895677, −8.802533810315117317666267545836, −8.098595130392906852147301827171, −6.69457680062517770050439432612, −5.71591959665765672153058519512, −5.43089290940691326561936030830, −3.60553558473410147214909858887, −2.72640701467110547273803256458, 0.28187624866950931376520413418, 2.46331325228814153072513003318, 3.79891057338873479951669688654, 4.47727728227798950793436009394, 5.90626350572469068537488291556, 6.97026772187840694660680321341, 7.63956641630494187382926434906, 9.270268707448885365737643739123, 9.930577262571662403912862677685, 10.93329777060508647463069405910

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