↑

Properties

Label 2-230-5.2-c4-0-9
Degree $2$
Conductor $230$
Sign $0.0227 - 0.999i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−2 − 2i)2-s + (−0.665 + 0.665i)3-s + 8i·4-s + (13.6 − 20.9i)5-s + 2.66·6-s + (18.0 + 18.0i)7-s + (16 − 16i)8-s + 80.1i·9-s + (−69.1 + 14.6i)10-s − 16.1·11-s + (−5.32 − 5.32i)12-s + (−160. + 160. i)13-s − 72.3i·14-s + (4.87 + 23.0i)15-s − 64·16-s + (−71.5 − 71.5i)17-s + ⋯
L(s)  = 1  + (−0.5 − 0.5i)2-s + (−0.0738 + 0.0738i)3-s + 0.5i·4-s + (0.544 − 0.838i)5-s + 0.0738·6-s + (0.369 + 0.369i)7-s + (0.250 − 0.250i)8-s + 0.989i·9-s + (−0.691 + 0.146i)10-s − 0.133·11-s + (−0.0369 − 0.0369i)12-s + (−0.948 + 0.948i)13-s − 0.369i·14-s + (0.0216 + 0.102i)15-s − 0.250·16-s + (−0.247 − 0.247i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0227 - 0.999i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.0227 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(230\)    =    \(2 \cdot 5 \cdot 23\)
Sign: $0.0227 - 0.999i$
Analytic conductor: \(23.7750\)
Root analytic conductor: \(4.87597\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{230} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 230,\ (\ :2),\ 0.0227 - 0.999i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.8151092809\)
\(L(\frac12)\) \(\approx\) \(0.8151092809\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2 + 2i)T \)
5 \( 1 + (-13.6 + 20.9i)T \)
23 \( 1 + (77.9 - 77.9i)T \)
good3 \( 1 + (0.665 - 0.665i)T - 81iT^{2} \)
7 \( 1 + (-18.0 - 18.0i)T + 2.40e3iT^{2} \)
11 \( 1 + 16.1T + 1.46e4T^{2} \)
13 \( 1 + (160. - 160. i)T - 2.85e4iT^{2} \)
17 \( 1 + (71.5 + 71.5i)T + 8.35e4iT^{2} \)
19 \( 1 + 344. iT - 1.30e5T^{2} \)
29 \( 1 - 1.24e3iT - 7.07e5T^{2} \)
31 \( 1 + 1.05e3T + 9.23e5T^{2} \)
37 \( 1 + (-1.24e3 - 1.24e3i)T + 1.87e6iT^{2} \)
41 \( 1 + 2.07e3T + 2.82e6T^{2} \)
43 \( 1 + (1.72e3 - 1.72e3i)T - 3.41e6iT^{2} \)
47 \( 1 + (-2.18e3 - 2.18e3i)T + 4.87e6iT^{2} \)
53 \( 1 + (2.81e3 - 2.81e3i)T - 7.89e6iT^{2} \)
59 \( 1 - 4.81e3iT - 1.21e7T^{2} \)
61 \( 1 - 1.53e3T + 1.38e7T^{2} \)
67 \( 1 + (638. + 638. i)T + 2.01e7iT^{2} \)
71 \( 1 - 9.22e3T + 2.54e7T^{2} \)
73 \( 1 + (-3.78e3 + 3.78e3i)T - 2.83e7iT^{2} \)
79 \( 1 - 1.20e4iT - 3.89e7T^{2} \)
83 \( 1 + (5.96e3 - 5.96e3i)T - 4.74e7iT^{2} \)
89 \( 1 + 1.11e4iT - 6.27e7T^{2} \)
97 \( 1 + (-1.47e3 - 1.47e3i)T + 8.85e7iT^{2} \)
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

−11.67675025041087640001597777774, −10.84387596687076167705240186943, −9.711163698634536413312928136998, −9.023064150302903355680955810888, −8.079389361633605078974348374120, −6.91577614486034726339500255813, −5.25327350448187358294059857446, −4.57413427764950273455923749038, −2.54699613958524573932684998383, −1.53365872223217627541873263176, 0.31615830346941132819624818230, 2.06235711688847736575079214490, 3.65676432324433908222109869155, 5.37488930657610833461869722831, 6.31763783186453944857573640187, 7.27910592173338446755573635518, 8.163432507637003615316237599901, 9.549476708180825965854091155068, 10.14402305858848376298916540988, 11.06532474369581856410916924765

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