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Properties

Label 2-230-115.114-c2-0-19
Degree $2$
Conductor $230$
Sign $0.0257 + 0.999i$
Analytic cond. $6.26704$
Root an. cond. $2.50340$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.41i·2-s − 3.00i·3-s − 2.00·4-s + (3.95 − 3.05i)5-s + 4.25·6-s − 7.53·7-s − 2.82i·8-s − 0.0375·9-s + (4.31 + 5.59i)10-s + 2.07i·11-s + 6.01i·12-s − 19.5i·13-s − 10.6i·14-s + (−9.18 − 11.9i)15-s + 4.00·16-s − 22.4·17-s + ⋯
L(s)  = 1  + 0.707i·2-s − 1.00i·3-s − 0.500·4-s + (0.791 − 0.610i)5-s + 0.708·6-s − 1.07·7-s − 0.353i·8-s − 0.00417·9-s + (0.431 + 0.559i)10-s + 0.189i·11-s + 0.501i·12-s − 1.50i·13-s − 0.761i·14-s + (−0.612 − 0.793i)15-s + 0.250·16-s − 1.31·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0257 + 0.999i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.0257 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(230\)    =    \(2 \cdot 5 \cdot 23\)
Sign: $0.0257 + 0.999i$
Analytic conductor: \(6.26704\)
Root analytic conductor: \(2.50340\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{230} (229, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 230,\ (\ :1),\ 0.0257 + 0.999i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.897854 - 0.875053i\)
\(L(\frac12)\) \(\approx\) \(0.897854 - 0.875053i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 1.41iT \)
5 \( 1 + (-3.95 + 3.05i)T \)
23 \( 1 + (18.5 + 13.5i)T \)
good3 \( 1 + 3.00iT - 9T^{2} \)
7 \( 1 + 7.53T + 49T^{2} \)
11 \( 1 - 2.07iT - 121T^{2} \)
13 \( 1 + 19.5iT - 169T^{2} \)
17 \( 1 + 22.4T + 289T^{2} \)
19 \( 1 + 7.65iT - 361T^{2} \)
29 \( 1 - 36.7T + 841T^{2} \)
31 \( 1 + 9.27T + 961T^{2} \)
37 \( 1 - 8.26T + 1.36e3T^{2} \)
41 \( 1 - 29.3T + 1.68e3T^{2} \)
43 \( 1 - 48.5T + 1.84e3T^{2} \)
47 \( 1 - 77.5iT - 2.20e3T^{2} \)
53 \( 1 - 39.4T + 2.80e3T^{2} \)
59 \( 1 + 19.3T + 3.48e3T^{2} \)
61 \( 1 + 11.9iT - 3.72e3T^{2} \)
67 \( 1 + 23.2T + 4.48e3T^{2} \)
71 \( 1 + 9.60T + 5.04e3T^{2} \)
73 \( 1 - 28.4iT - 5.32e3T^{2} \)
79 \( 1 - 88.7iT - 6.24e3T^{2} \)
83 \( 1 - 70.8T + 6.88e3T^{2} \)
89 \( 1 + 130. iT - 7.92e3T^{2} \)
97 \( 1 - 169.T + 9.40e3T^{2} \)
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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

−12.34995945235809208877044201356, −10.53136844972825719195622877355, −9.636741224134554441203985466165, −8.644791856084205800516513511356, −7.62134565796695192728555803177, −6.51308358597456688297527742319, −5.95529018139115055903238665694, −4.52992057731903793075029990082, −2.55400998704332161844517753946, −0.67228152665357585385279443463, 2.15184616569000785772427900700, 3.53649000667236811769382691657, 4.50459730674665409719567495340, 6.01583224481104252523477665117, 6.96542491738954440924292042662, 8.942739357825122235479904923517, 9.520781439720731365118358683944, 10.23668359458833629928292369975, 10.99580235293286170757341696806, 12.05756935702375912792356143074

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