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Properties

Label 2-230-115.114-c4-0-27
Degree $2$
Conductor $230$
Sign $0.834 - 0.550i$
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  + 2.82i·2-s + 6.09i·3-s − 8.00·4-s + (−24.9 + 0.899i)5-s − 17.2·6-s + 17.6·7-s − 22.6i·8-s + 43.8·9-s + (−2.54 − 70.6i)10-s − 123. i·11-s − 48.7i·12-s − 72.7i·13-s + 49.8i·14-s + (−5.48 − 152. i)15-s + 64.0·16-s − 134.·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + 0.677i·3-s − 0.500·4-s + (−0.999 + 0.0359i)5-s − 0.479·6-s + 0.359·7-s − 0.353i·8-s + 0.541·9-s + (−0.0254 − 0.706i)10-s − 1.02i·11-s − 0.338i·12-s − 0.430i·13-s + 0.254i·14-s + (−0.0243 − 0.677i)15-s + 0.250·16-s − 0.465·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.834 - 0.550i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.834 - 0.550i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(230\)    =    \(2 \cdot 5 \cdot 23\)
Sign: $0.834 - 0.550i$
Analytic conductor: \(23.7750\)
Root analytic conductor: \(4.87597\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{230} (229, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 230,\ (\ :2),\ 0.834 - 0.550i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.468177346\)
\(L(\frac12)\) \(\approx\) \(1.468177346\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 2.82iT \)
5 \( 1 + (24.9 - 0.899i)T \)
23 \( 1 + (-275. - 451. i)T \)
good3 \( 1 - 6.09iT - 81T^{2} \)
7 \( 1 - 17.6T + 2.40e3T^{2} \)
11 \( 1 + 123. iT - 1.46e4T^{2} \)
13 \( 1 + 72.7iT - 2.85e4T^{2} \)
17 \( 1 + 134.T + 8.35e4T^{2} \)
19 \( 1 + 473. iT - 1.30e5T^{2} \)
29 \( 1 - 724.T + 7.07e5T^{2} \)
31 \( 1 + 464.T + 9.23e5T^{2} \)
37 \( 1 - 1.37e3T + 1.87e6T^{2} \)
41 \( 1 - 403.T + 2.82e6T^{2} \)
43 \( 1 - 722.T + 3.41e6T^{2} \)
47 \( 1 + 580. iT - 4.87e6T^{2} \)
53 \( 1 - 1.49e3T + 7.89e6T^{2} \)
59 \( 1 + 1.20e3T + 1.21e7T^{2} \)
61 \( 1 + 1.99e3iT - 1.38e7T^{2} \)
67 \( 1 - 4.84e3T + 2.01e7T^{2} \)
71 \( 1 - 9.04e3T + 2.54e7T^{2} \)
73 \( 1 + 2.91e3iT - 2.83e7T^{2} \)
79 \( 1 + 3.11e3iT - 3.89e7T^{2} \)
83 \( 1 - 196.T + 4.74e7T^{2} \)
89 \( 1 + 4.89e3iT - 6.27e7T^{2} \)
97 \( 1 + 117.T + 8.85e7T^{2} \)
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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

−11.35615318775985013616907751508, −10.82029040836583520618525105530, −9.485702375013851588586833927258, −8.593360523943613884952702178099, −7.66984193731508596389815415351, −6.69099806853569696455960860924, −5.21881462096022456202797694789, −4.34782394759911732026104174121, −3.23429924677270240460742059111, −0.65832203861746150102739877371, 1.03031266984969372452970133957, 2.27014633940985671801270122929, 3.95927298681662302842372190625, 4.75789084090515954867742927804, 6.59406487891456478352133798583, 7.56977397761424444133094339408, 8.377464882028463341955782212720, 9.636110284800878235129554651503, 10.66255955623422398889526780702, 11.62292968069407919831300434693

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