↑

Properties

Label 2-230-5.2-c2-0-20
Degree $2$
Conductor $230$
Sign $0.494 + 0.869i$
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 + (2.20 − 2.20i)3-s + 2i·4-s + (0.183 − 4.99i)5-s + 4.40·6-s + (−8.35 − 8.35i)7-s + (−2 + 2i)8-s − 0.680i·9-s + (5.18 − 4.81i)10-s + 4.85·11-s + (4.40 + 4.40i)12-s + (17.1 − 17.1i)13-s − 16.7i·14-s + (−10.5 − 11.3i)15-s − 4·16-s + (−9.16 − 9.16i)17-s + ⋯
L(s)  = 1  + (0.5 + 0.5i)2-s + (0.733 − 0.733i)3-s + 0.5i·4-s + (0.0367 − 0.999i)5-s + 0.733·6-s + (−1.19 − 1.19i)7-s + (−0.250 + 0.250i)8-s − 0.0756i·9-s + (0.518 − 0.481i)10-s + 0.441·11-s + (0.366 + 0.366i)12-s + (1.32 − 1.32i)13-s − 1.19i·14-s + (−0.705 − 0.759i)15-s − 0.250·16-s + (−0.538 − 0.538i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.494 + 0.869i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.494 + 0.869i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(230\)    =    \(2 \cdot 5 \cdot 23\)
Sign: $0.494 + 0.869i$
Analytic conductor: \(6.26704\)
Root analytic conductor: \(2.50340\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{230} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 230,\ (\ :1),\ 0.494 + 0.869i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.96777 - 1.14497i\)
\(L(\frac12)\) \(\approx\) \(1.96777 - 1.14497i\)
\(L(2)\) 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 \( 1 + (-0.183 + 4.99i)T \)
23 \( 1 + (-3.39 + 3.39i)T \)
good3 \( 1 + (-2.20 + 2.20i)T - 9iT^{2} \)
7 \( 1 + (8.35 + 8.35i)T + 49iT^{2} \)
11 \( 1 - 4.85T + 121T^{2} \)
13 \( 1 + (-17.1 + 17.1i)T - 169iT^{2} \)
17 \( 1 + (9.16 + 9.16i)T + 289iT^{2} \)
19 \( 1 - 28.3iT - 361T^{2} \)
29 \( 1 - 16.7iT - 841T^{2} \)
31 \( 1 - 49.9T + 961T^{2} \)
37 \( 1 + (15.0 + 15.0i)T + 1.36e3iT^{2} \)
41 \( 1 - 43.9T + 1.68e3T^{2} \)
43 \( 1 + (5.23 - 5.23i)T - 1.84e3iT^{2} \)
47 \( 1 + (-20.2 - 20.2i)T + 2.20e3iT^{2} \)
53 \( 1 + (34.0 - 34.0i)T - 2.80e3iT^{2} \)
59 \( 1 + 67.5iT - 3.48e3T^{2} \)
61 \( 1 + 10.1T + 3.72e3T^{2} \)
67 \( 1 + (-18.0 - 18.0i)T + 4.48e3iT^{2} \)
71 \( 1 - 21.4T + 5.04e3T^{2} \)
73 \( 1 + (21.4 - 21.4i)T - 5.32e3iT^{2} \)
79 \( 1 - 113. iT - 6.24e3T^{2} \)
83 \( 1 + (45.6 - 45.6i)T - 6.88e3iT^{2} \)
89 \( 1 - 64.2iT - 7.92e3T^{2} \)
97 \( 1 + (98.3 + 98.3i)T + 9.40e3iT^{2} \)
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

−12.52180459557469572207922053182, −10.85183622378365838293942436045, −9.699432730373960441758814604351, −8.501866937904742695389223203367, −7.86150685595382595220277012388, −6.77759354681984208821062731478, −5.77559162411590781018396701332, −4.19912528916000362895756497280, −3.16556355810644575196519521859, −1.05283907216392655886422230615, 2.43660738001541820037376869231, 3.32104218938399515332167885847, 4.28228849999623214369950720333, 6.20524733406188394335387037869, 6.59690932511793735054891481680, 8.779593853406271906867306797362, 9.234246328391852102243392957963, 10.14809095223063939007612875223, 11.27689025096083318359372119629, 11.96806402833933879935176065320

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