Properties

Label 2-230-23.22-c2-0-12
Degree $2$
Conductor $230$
Sign $0.993 + 0.112i$
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  + 1.41·2-s + 4.30·3-s + 2.00·4-s − 2.23i·5-s + 6.09·6-s + 1.47i·7-s + 2.82·8-s + 9.55·9-s − 3.16i·10-s + 6.04i·11-s + 8.61·12-s − 5.21·13-s + 2.08i·14-s − 9.63i·15-s + 4.00·16-s − 15.7i·17-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.43·3-s + 0.500·4-s − 0.447i·5-s + 1.01·6-s + 0.210i·7-s + 0.353·8-s + 1.06·9-s − 0.316i·10-s + 0.549i·11-s + 0.717·12-s − 0.401·13-s + 0.149i·14-s − 0.642i·15-s + 0.250·16-s − 0.923i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 + 0.112i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.993 + 0.112i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(230\)    =    \(2 \cdot 5 \cdot 23\)
Sign: $0.993 + 0.112i$
Analytic conductor: \(6.26704\)
Root analytic conductor: \(2.50340\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{230} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 230,\ (\ :1),\ 0.993 + 0.112i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(3.50319 - 0.197115i\)
\(L(\frac12)\) \(\approx\) \(3.50319 - 0.197115i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 1.41T \)
5 \( 1 + 2.23iT \)
23 \( 1 + (2.58 - 22.8i)T \)
good3 \( 1 - 4.30T + 9T^{2} \)
7 \( 1 - 1.47iT - 49T^{2} \)
11 \( 1 - 6.04iT - 121T^{2} \)
13 \( 1 + 5.21T + 169T^{2} \)
17 \( 1 + 15.7iT - 289T^{2} \)
19 \( 1 + 4.82iT - 361T^{2} \)
29 \( 1 + 23.4T + 841T^{2} \)
31 \( 1 + 20.4T + 961T^{2} \)
37 \( 1 + 15.5iT - 1.36e3T^{2} \)
41 \( 1 - 20.3T + 1.68e3T^{2} \)
43 \( 1 - 38.1iT - 1.84e3T^{2} \)
47 \( 1 + 13.8T + 2.20e3T^{2} \)
53 \( 1 - 38.2iT - 2.80e3T^{2} \)
59 \( 1 + 33.5T + 3.48e3T^{2} \)
61 \( 1 + 100. iT - 3.72e3T^{2} \)
67 \( 1 - 32.4iT - 4.48e3T^{2} \)
71 \( 1 + 24.1T + 5.04e3T^{2} \)
73 \( 1 - 15.1T + 5.32e3T^{2} \)
79 \( 1 - 11.2iT - 6.24e3T^{2} \)
83 \( 1 + 44.1iT - 6.88e3T^{2} \)
89 \( 1 - 111. iT - 7.92e3T^{2} \)
97 \( 1 + 154. iT - 9.40e3T^{2} \)
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

−12.25928599556959807932678697201, −11.18504609871986250428365107913, −9.670712369447652528099194898927, −9.138347776505082470232370047812, −7.88250427250018570859625836376, −7.16783440963346792715455669187, −5.52483753706758343949500359983, −4.34194170568847176470301718332, −3.13983258056870335695236089862, −1.98484653869137568546405023694, 2.10531419447362216751149016206, 3.25048400981148179439600801926, 4.15757545772656088643431420066, 5.80591458399688865510423525226, 7.06178841392708776897231137193, 8.005745351160533964688404349985, 8.911726367472106945876055037027, 10.11001031318905869074194469366, 10.99500616372114808632480950924, 12.26416039428599729892952213699

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