Properties

Label 2-230-23.3-c1-0-6
Degree $2$
Conductor $230$
Sign $0.710 + 0.703i$
Analytic cond. $1.83655$
Root an. cond. $1.35519$
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  + (0.142 + 0.989i)2-s + (1.40 − 3.07i)3-s + (−0.959 + 0.281i)4-s + (−0.654 + 0.755i)5-s + (3.24 + 0.953i)6-s + (2.97 − 1.91i)7-s + (−0.415 − 0.909i)8-s + (−5.53 − 6.38i)9-s + (−0.841 − 0.540i)10-s + (−0.474 + 3.30i)11-s + (−0.481 + 3.34i)12-s + (−2.04 − 1.31i)13-s + (2.31 + 2.67i)14-s + (1.40 + 3.07i)15-s + (0.841 − 0.540i)16-s + (4.17 + 1.22i)17-s + ⋯
L(s)  = 1  + (0.100 + 0.699i)2-s + (0.811 − 1.77i)3-s + (−0.479 + 0.140i)4-s + (−0.292 + 0.337i)5-s + (1.32 + 0.389i)6-s + (1.12 − 0.722i)7-s + (−0.146 − 0.321i)8-s + (−1.84 − 2.12i)9-s + (−0.266 − 0.170i)10-s + (−0.143 + 0.995i)11-s + (−0.138 + 0.966i)12-s + (−0.567 − 0.364i)13-s + (0.619 + 0.714i)14-s + (0.362 + 0.794i)15-s + (0.210 − 0.135i)16-s + (1.01 + 0.297i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.710 + 0.703i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.710 + 0.703i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(230\)    =    \(2 \cdot 5 \cdot 23\)
Sign: $0.710 + 0.703i$
Analytic conductor: \(1.83655\)
Root analytic conductor: \(1.35519\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{230} (141, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 230,\ (\ :1/2),\ 0.710 + 0.703i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.44017 - 0.592681i\)
\(L(\frac12)\) \(\approx\) \(1.44017 - 0.592681i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.142 - 0.989i)T \)
5 \( 1 + (0.654 - 0.755i)T \)
23 \( 1 + (-4.72 - 0.824i)T \)
good3 \( 1 + (-1.40 + 3.07i)T + (-1.96 - 2.26i)T^{2} \)
7 \( 1 + (-2.97 + 1.91i)T + (2.90 - 6.36i)T^{2} \)
11 \( 1 + (0.474 - 3.30i)T + (-10.5 - 3.09i)T^{2} \)
13 \( 1 + (2.04 + 1.31i)T + (5.40 + 11.8i)T^{2} \)
17 \( 1 + (-4.17 - 1.22i)T + (14.3 + 9.19i)T^{2} \)
19 \( 1 + (1.70 - 0.501i)T + (15.9 - 10.2i)T^{2} \)
29 \( 1 + (-4.26 - 1.25i)T + (24.3 + 15.6i)T^{2} \)
31 \( 1 + (-0.962 - 2.10i)T + (-20.3 + 23.4i)T^{2} \)
37 \( 1 + (-2.47 - 2.85i)T + (-5.26 + 36.6i)T^{2} \)
41 \( 1 + (5.45 - 6.29i)T + (-5.83 - 40.5i)T^{2} \)
43 \( 1 + (1.86 - 4.08i)T + (-28.1 - 32.4i)T^{2} \)
47 \( 1 + 6.54T + 47T^{2} \)
53 \( 1 + (6.75 - 4.34i)T + (22.0 - 48.2i)T^{2} \)
59 \( 1 + (-11.7 - 7.52i)T + (24.5 + 53.6i)T^{2} \)
61 \( 1 + (4.41 + 9.67i)T + (-39.9 + 46.1i)T^{2} \)
67 \( 1 + (-0.522 - 3.63i)T + (-64.2 + 18.8i)T^{2} \)
71 \( 1 + (1.28 + 8.93i)T + (-68.1 + 20.0i)T^{2} \)
73 \( 1 + (2.35 - 0.690i)T + (61.4 - 39.4i)T^{2} \)
79 \( 1 + (1.93 + 1.24i)T + (32.8 + 71.8i)T^{2} \)
83 \( 1 + (-0.921 - 1.06i)T + (-11.8 + 82.1i)T^{2} \)
89 \( 1 + (-6.81 + 14.9i)T + (-58.2 - 67.2i)T^{2} \)
97 \( 1 + (-7.51 + 8.67i)T + (-13.8 - 96.0i)T^{2} \)
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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

−12.41044461831501988485517211572, −11.47965244028006394513506851159, −9.988997626910177938434944356409, −8.528465909087020135810511168075, −7.79176660189252116068307216140, −7.33123357412734488321884620412, −6.42040002877288213193430377916, −4.82774356682249185156939252909, −3.09094388812801954353108935899, −1.43016499528759867903834316466, 2.50156362145436638738342834881, 3.64328212554166266657907772934, 4.82560711386507855592890595746, 5.36791140507418324696648285215, 8.071618598199080597232123016020, 8.626487514775625061165614734181, 9.433684937043496193363129435842, 10.41701982352004998044162185812, 11.25697789675465184254199859211, 11.91833916918355807298401358247

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