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Properties

Label 2-230-23.22-c4-0-5
Degree $2$
Conductor $230$
Sign $-0.948 + 0.316i$
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  − 2.82·2-s − 5.93·3-s + 8.00·4-s + 11.1i·5-s + 16.7·6-s + 85.3i·7-s − 22.6·8-s − 45.7·9-s − 31.6i·10-s + 149. i·11-s − 47.5·12-s + 94.2·13-s − 241. i·14-s − 66.4i·15-s + 64.0·16-s − 334. i·17-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.659·3-s + 0.500·4-s + 0.447i·5-s + 0.466·6-s + 1.74i·7-s − 0.353·8-s − 0.564·9-s − 0.316i·10-s + 1.23i·11-s − 0.329·12-s + 0.557·13-s − 1.23i·14-s − 0.295i·15-s + 0.250·16-s − 1.15i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.948 + 0.316i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.948 + 0.316i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(230\)    =    \(2 \cdot 5 \cdot 23\)
Sign: $-0.948 + 0.316i$
Analytic conductor: \(23.7750\)
Root analytic conductor: \(4.87597\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{230} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 230,\ (\ :2),\ -0.948 + 0.316i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.5137690104\)
\(L(\frac12)\) \(\approx\) \(0.5137690104\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 2.82T \)
5 \( 1 - 11.1iT \)
23 \( 1 + (-167. - 501. i)T \)
good3 \( 1 + 5.93T + 81T^{2} \)
7 \( 1 - 85.3iT - 2.40e3T^{2} \)
11 \( 1 - 149. iT - 1.46e4T^{2} \)
13 \( 1 - 94.2T + 2.85e4T^{2} \)
17 \( 1 + 334. iT - 8.35e4T^{2} \)
19 \( 1 - 531. iT - 1.30e5T^{2} \)
29 \( 1 - 1.63e3T + 7.07e5T^{2} \)
31 \( 1 + 1.72e3T + 9.23e5T^{2} \)
37 \( 1 + 102. iT - 1.87e6T^{2} \)
41 \( 1 + 1.85e3T + 2.82e6T^{2} \)
43 \( 1 + 1.44e3iT - 3.41e6T^{2} \)
47 \( 1 + 746.T + 4.87e6T^{2} \)
53 \( 1 - 398. iT - 7.89e6T^{2} \)
59 \( 1 + 1.77e3T + 1.21e7T^{2} \)
61 \( 1 - 351. iT - 1.38e7T^{2} \)
67 \( 1 + 4.94e3iT - 2.01e7T^{2} \)
71 \( 1 - 115.T + 2.54e7T^{2} \)
73 \( 1 + 6.25e3T + 2.83e7T^{2} \)
79 \( 1 - 1.00e4iT - 3.89e7T^{2} \)
83 \( 1 - 1.70e3iT - 4.74e7T^{2} \)
89 \( 1 - 3.51e3iT - 6.27e7T^{2} \)
97 \( 1 + 1.06e4iT - 8.85e7T^{2} \)
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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

−11.97673276013212808264489227616, −11.20117881655285992162377965429, −10.07812857820182518321772750810, −9.189938979452063685012351714262, −8.282846498426775673848399418831, −7.02948433719649119147862507383, −5.96210421822236054571329200917, −5.17719260928112276376200759727, −3.05881270158069044330864899436, −1.83205960335145053353026363999, 0.28052335642262504845873892699, 1.08040296249527358042465802918, 3.28444020846415017684346350778, 4.67130342640560126838963746922, 6.08848136511258330954873217708, 6.89877377414651434187017227205, 8.224951041780207319817383500782, 8.861783136301342970125488912301, 10.41082127241185996048766301453, 10.83139716258838808055031294357

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