↑

Properties

Label 2-230-5.3-c2-0-20
Degree $2$
Conductor $230$
Sign $-0.945 - 0.325i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (1 − i)2-s + (0.561 + 0.561i)3-s − 2i·4-s + (−4.87 + 1.10i)5-s + 1.12·6-s + (−8.38 + 8.38i)7-s + (−2 − 2i)8-s − 8.36i·9-s + (−3.77 + 5.97i)10-s − 11.2·11-s + (1.12 − 1.12i)12-s + (−15.9 − 15.9i)13-s + 16.7i·14-s + (−3.35 − 2.12i)15-s − 4·16-s + (14.2 − 14.2i)17-s + ⋯
L(s)  = 1  + (0.5 − 0.5i)2-s + (0.187 + 0.187i)3-s − 0.5i·4-s + (−0.975 + 0.220i)5-s + 0.187·6-s + (−1.19 + 1.19i)7-s + (−0.250 − 0.250i)8-s − 0.929i·9-s + (−0.377 + 0.597i)10-s − 1.02·11-s + (0.0936 − 0.0936i)12-s + (−1.22 − 1.22i)13-s + 1.19i·14-s + (−0.223 − 0.141i)15-s − 0.250·16-s + (0.840 − 0.840i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.945 - 0.325i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.945 - 0.325i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(230\)    =    \(2 \cdot 5 \cdot 23\)
Sign: $-0.945 - 0.325i$
Analytic conductor: \(6.26704\)
Root analytic conductor: \(2.50340\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{230} (93, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 230,\ (\ :1),\ -0.945 - 0.325i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0148753 + 0.0889007i\)
\(L(\frac12)\) \(\approx\) \(0.0148753 + 0.0889007i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1 + i)T \)
5 \( 1 + (4.87 - 1.10i)T \)
23 \( 1 + (3.39 + 3.39i)T \)
good3 \( 1 + (-0.561 - 0.561i)T + 9iT^{2} \)
7 \( 1 + (8.38 - 8.38i)T - 49iT^{2} \)
11 \( 1 + 11.2T + 121T^{2} \)
13 \( 1 + (15.9 + 15.9i)T + 169iT^{2} \)
17 \( 1 + (-14.2 + 14.2i)T - 289iT^{2} \)
19 \( 1 - 35.4iT - 361T^{2} \)
29 \( 1 - 26.2iT - 841T^{2} \)
31 \( 1 - 19.1T + 961T^{2} \)
37 \( 1 + (26.7 - 26.7i)T - 1.36e3iT^{2} \)
41 \( 1 - 19.3T + 1.68e3T^{2} \)
43 \( 1 + (44.0 + 44.0i)T + 1.84e3iT^{2} \)
47 \( 1 + (36.1 - 36.1i)T - 2.20e3iT^{2} \)
53 \( 1 + (0.502 + 0.502i)T + 2.80e3iT^{2} \)
59 \( 1 - 5.81iT - 3.48e3T^{2} \)
61 \( 1 + 0.344T + 3.72e3T^{2} \)
67 \( 1 + (17.5 - 17.5i)T - 4.48e3iT^{2} \)
71 \( 1 + 14.8T + 5.04e3T^{2} \)
73 \( 1 + (68.0 + 68.0i)T + 5.32e3iT^{2} \)
79 \( 1 + 82.9iT - 6.24e3T^{2} \)
83 \( 1 + (32.8 + 32.8i)T + 6.88e3iT^{2} \)
89 \( 1 + 59.4iT - 7.92e3T^{2} \)
97 \( 1 + (-33.5 + 33.5i)T - 9.40e3iT^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.99488889679371719923127745046, −10.26152477450190125952845385724, −9.870321015291401879748350748111, −8.551904600446263072977418404470, −7.43536412200146984708487513847, −6.06094193287075217419082509231, −5.07743375286579831775219726416, −3.32950847649151622376215845867, −2.93014369341903769126145044156, −0.03775550455115702524712451396, 2.79993247377135774267530920402, 4.13599923163581377512106852346, 5.01991900610686310724264124227, 6.77689474009757637325577532055, 7.37145503520473435564835877649, 8.188519958403216291401160126071, 9.610964827052234128488353087276, 10.63563324630596982367373468448, 11.71684263469988121918964672029, 12.83745647025577681488101531181

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