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Properties

Label 2-430-43.16-c1-0-11
Degree $2$
Conductor $430$
Sign $-0.704 + 0.709i$
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  + (0.222 + 0.974i)2-s + (0.697 − 3.05i)3-s + (−0.900 + 0.433i)4-s + (−0.623 − 0.781i)5-s + 3.13·6-s − 2.56·7-s + (−0.623 − 0.781i)8-s + (−6.15 − 2.96i)9-s + (0.623 − 0.781i)10-s + (−1.56 − 0.754i)11-s + (0.697 + 3.05i)12-s + (−0.536 − 0.672i)13-s + (−0.571 − 2.50i)14-s + (−2.82 + 1.36i)15-s + (0.623 − 0.781i)16-s + (−1.12 + 1.41i)17-s + ⋯
L(s)  = 1  + (0.157 + 0.689i)2-s + (0.402 − 1.76i)3-s + (−0.450 + 0.216i)4-s + (−0.278 − 0.349i)5-s + 1.27·6-s − 0.970·7-s + (−0.220 − 0.276i)8-s + (−2.05 − 0.987i)9-s + (0.197 − 0.247i)10-s + (−0.472 − 0.227i)11-s + (0.201 + 0.882i)12-s + (−0.148 − 0.186i)13-s + (−0.152 − 0.668i)14-s + (−0.729 + 0.351i)15-s + (0.155 − 0.195i)16-s + (−0.273 + 0.343i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.704 + 0.709i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.704 + 0.709i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(430\)    =    \(2 \cdot 5 \cdot 43\)
Sign: $-0.704 + 0.709i$
Analytic conductor: \(3.43356\)
Root analytic conductor: \(1.85298\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{430} (231, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 430,\ (\ :1/2),\ -0.704 + 0.709i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.359597 - 0.863235i\)
\(L(\frac12)\) \(\approx\) \(0.359597 - 0.863235i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.222 - 0.974i)T \)
5 \( 1 + (0.623 + 0.781i)T \)
43 \( 1 + (-4.77 + 4.49i)T \)
good3 \( 1 + (-0.697 + 3.05i)T + (-2.70 - 1.30i)T^{2} \)
7 \( 1 + 2.56T + 7T^{2} \)
11 \( 1 + (1.56 + 0.754i)T + (6.85 + 8.60i)T^{2} \)
13 \( 1 + (0.536 + 0.672i)T + (-2.89 + 12.6i)T^{2} \)
17 \( 1 + (1.12 - 1.41i)T + (-3.78 - 16.5i)T^{2} \)
19 \( 1 + (-0.880 + 0.424i)T + (11.8 - 14.8i)T^{2} \)
23 \( 1 + (-5.19 - 2.50i)T + (14.3 + 17.9i)T^{2} \)
29 \( 1 + (1.38 + 6.05i)T + (-26.1 + 12.5i)T^{2} \)
31 \( 1 + (1.62 + 7.12i)T + (-27.9 + 13.4i)T^{2} \)
37 \( 1 - 0.427T + 37T^{2} \)
41 \( 1 + (2.27 + 9.98i)T + (-36.9 + 17.7i)T^{2} \)
47 \( 1 + (-7.48 + 3.60i)T + (29.3 - 36.7i)T^{2} \)
53 \( 1 + (1.83 - 2.30i)T + (-11.7 - 51.6i)T^{2} \)
59 \( 1 + (8.84 - 11.0i)T + (-13.1 - 57.5i)T^{2} \)
61 \( 1 + (-1.01 + 4.46i)T + (-54.9 - 26.4i)T^{2} \)
67 \( 1 + (-6.22 + 2.99i)T + (41.7 - 52.3i)T^{2} \)
71 \( 1 + (-11.3 + 5.46i)T + (44.2 - 55.5i)T^{2} \)
73 \( 1 + (-1.87 - 2.35i)T + (-16.2 + 71.1i)T^{2} \)
79 \( 1 - 12.0T + 79T^{2} \)
83 \( 1 + (2.38 - 10.4i)T + (-74.7 - 36.0i)T^{2} \)
89 \( 1 + (-2.00 + 8.79i)T + (-80.1 - 38.6i)T^{2} \)
97 \( 1 + (13.9 + 6.72i)T + (60.4 + 75.8i)T^{2} \)
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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

−11.05883337220057738083180200487, −9.491142309216663457754632524737, −8.711006943231302265891011658170, −7.76829874951159796211049871432, −7.22020179832728827596470725713, −6.28545937424379708645552752885, −5.48484236916119902352738064212, −3.67567630922611755505121432373, −2.42675513550738999800664590589, −0.52010588630604205935520717313, 2.81439781052472505651210851713, 3.37086994081330203922175470579, 4.50544451828840487697764685453, 5.28870161037533001836704662328, 6.74992443753902081935030987737, 8.241807432368720208933460087961, 9.317595881690773682998186857593, 9.638389619085175735675926202344, 10.75040151759293926810487609069, 10.95597681787579107667268377347

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