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Properties

Label 2-230-5.3-c4-0-10
Degree $2$
Conductor $230$
Sign $-0.425 - 0.905i$
Analytic cond. $23.7750$
Root an. cond. $4.87597$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2 + 2i)2-s + (4.47 + 4.47i)3-s − 8i·4-s + (−20.9 − 13.6i)5-s − 17.9·6-s + (−15.4 + 15.4i)7-s + (16 + 16i)8-s − 40.9i·9-s + (69.1 − 14.5i)10-s + 133.·11-s + (35.8 − 35.8i)12-s + (−27.8 − 27.8i)13-s − 61.6i·14-s + (−32.6 − 154. i)15-s − 64·16-s + (−6.97 + 6.97i)17-s + ⋯
L(s)  = 1  + (−0.5 + 0.5i)2-s + (0.497 + 0.497i)3-s − 0.5i·4-s + (−0.837 − 0.546i)5-s − 0.497·6-s + (−0.314 + 0.314i)7-s + (0.250 + 0.250i)8-s − 0.505i·9-s + (0.691 − 0.145i)10-s + 1.10·11-s + (0.248 − 0.248i)12-s + (−0.164 − 0.164i)13-s − 0.314i·14-s + (−0.144 − 0.688i)15-s − 0.250·16-s + (−0.0241 + 0.0241i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.425 - 0.905i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.425 - 0.905i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(230\)    =    \(2 \cdot 5 \cdot 23\)
Sign: $-0.425 - 0.905i$
Analytic conductor: \(23.7750\)
Root analytic conductor: \(4.87597\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{230} (93, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 230,\ (\ :2),\ -0.425 - 0.905i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.131050109\)
\(L(\frac12)\) \(\approx\) \(1.131050109\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2 - 2i)T \)
5 \( 1 + (20.9 + 13.6i)T \)
23 \( 1 + (77.9 + 77.9i)T \)
good3 \( 1 + (-4.47 - 4.47i)T + 81iT^{2} \)
7 \( 1 + (15.4 - 15.4i)T - 2.40e3iT^{2} \)
11 \( 1 - 133.T + 1.46e4T^{2} \)
13 \( 1 + (27.8 + 27.8i)T + 2.85e4iT^{2} \)
17 \( 1 + (6.97 - 6.97i)T - 8.35e4iT^{2} \)
19 \( 1 - 581. iT - 1.30e5T^{2} \)
29 \( 1 - 1.42e3iT - 7.07e5T^{2} \)
31 \( 1 + 109.T + 9.23e5T^{2} \)
37 \( 1 + (-347. + 347. i)T - 1.87e6iT^{2} \)
41 \( 1 + 1.50e3T + 2.82e6T^{2} \)
43 \( 1 + (-687. - 687. i)T + 3.41e6iT^{2} \)
47 \( 1 + (97.5 - 97.5i)T - 4.87e6iT^{2} \)
53 \( 1 + (-2.45e3 - 2.45e3i)T + 7.89e6iT^{2} \)
59 \( 1 - 3.82e3iT - 1.21e7T^{2} \)
61 \( 1 + 5.06e3T + 1.38e7T^{2} \)
67 \( 1 + (-1.52e3 + 1.52e3i)T - 2.01e7iT^{2} \)
71 \( 1 + 1.39e3T + 2.54e7T^{2} \)
73 \( 1 + (-1.26e3 - 1.26e3i)T + 2.83e7iT^{2} \)
79 \( 1 - 5.01e3iT - 3.89e7T^{2} \)
83 \( 1 + (-3.63e3 - 3.63e3i)T + 4.74e7iT^{2} \)
89 \( 1 - 1.20e4iT - 6.27e7T^{2} \)
97 \( 1 + (-897. + 897. i)T - 8.85e7iT^{2} \)
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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

−12.01171611985240638848189021795, −10.67887385030157236836554251560, −9.542757815016424555815316484941, −8.931050721602264755626400832594, −8.123170954286697493097136475006, −6.95453093905827029851665641755, −5.79703195944008084739306185867, −4.32879380425462525055983669955, −3.37079968387925174873006321511, −1.21030217345819372429041356310, 0.48385058142177818348779694647, 2.15220961440813012470951279204, 3.33684251682388167726548816996, 4.49796739391627495625748908212, 6.62305144384129099528706544055, 7.33147259096541957793351434136, 8.267499212486218137026109353054, 9.220828611699347294294704165255, 10.32164318127180136413120400973, 11.35274327566129469898794737228

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