↑

Properties

Label 2-230-5.2-c4-0-7
Degree $2$
Conductor $230$
Sign $-0.832 - 0.554i$
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 + (3.59 − 3.59i)3-s + 8i·4-s + (−10.4 − 22.7i)5-s + 14.3·6-s + (7.45 + 7.45i)7-s + (−16 + 16i)8-s + 55.0i·9-s + (24.6 − 66.2i)10-s − 221.·11-s + (28.7 + 28.7i)12-s + (−43.7 + 43.7i)13-s + 29.8i·14-s + (−119. − 44.2i)15-s − 64·16-s + (192. + 192. i)17-s + ⋯
L(s)  = 1  + (0.5 + 0.5i)2-s + (0.399 − 0.399i)3-s + 0.5i·4-s + (−0.416 − 0.909i)5-s + 0.399·6-s + (0.152 + 0.152i)7-s + (−0.250 + 0.250i)8-s + 0.680i·9-s + (0.246 − 0.662i)10-s − 1.83·11-s + (0.199 + 0.199i)12-s + (−0.258 + 0.258i)13-s + 0.152i·14-s + (−0.530 − 0.196i)15-s − 0.250·16-s + (0.667 + 0.667i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.832 - 0.554i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.832 - 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(230\)    =    \(2 \cdot 5 \cdot 23\)
Sign: $-0.832 - 0.554i$
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: \((2,\ 230,\ (\ :2),\ -0.832 - 0.554i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.091992504\)
\(L(\frac12)\) \(\approx\) \(1.091992504\)
\(L(3)\) 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 \( 1 + (10.4 + 22.7i)T \)
23 \( 1 + (-77.9 + 77.9i)T \)
good3 \( 1 + (-3.59 + 3.59i)T - 81iT^{2} \)
7 \( 1 + (-7.45 - 7.45i)T + 2.40e3iT^{2} \)
11 \( 1 + 221.T + 1.46e4T^{2} \)
13 \( 1 + (43.7 - 43.7i)T - 2.85e4iT^{2} \)
17 \( 1 + (-192. - 192. i)T + 8.35e4iT^{2} \)
19 \( 1 - 228. iT - 1.30e5T^{2} \)
29 \( 1 - 1.15e3iT - 7.07e5T^{2} \)
31 \( 1 + 731.T + 9.23e5T^{2} \)
37 \( 1 + (-941. - 941. i)T + 1.87e6iT^{2} \)
41 \( 1 + 2.64e3T + 2.82e6T^{2} \)
43 \( 1 + (354. - 354. i)T - 3.41e6iT^{2} \)
47 \( 1 + (1.87e3 + 1.87e3i)T + 4.87e6iT^{2} \)
53 \( 1 + (560. - 560. i)T - 7.89e6iT^{2} \)
59 \( 1 + 5.55e3iT - 1.21e7T^{2} \)
61 \( 1 + 18.0T + 1.38e7T^{2} \)
67 \( 1 + (-1.14e3 - 1.14e3i)T + 2.01e7iT^{2} \)
71 \( 1 + 1.58e3T + 2.54e7T^{2} \)
73 \( 1 + (-3.93e3 + 3.93e3i)T - 2.83e7iT^{2} \)
79 \( 1 - 9.83e3iT - 3.89e7T^{2} \)
83 \( 1 + (4.21e3 - 4.21e3i)T - 4.74e7iT^{2} \)
89 \( 1 + 39.1iT - 6.27e7T^{2} \)
97 \( 1 + (9.31e3 + 9.31e3i)T + 8.85e7iT^{2} \)
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

−12.38906374217684049456390437807, −11.10700018226452168302162775277, −9.975274621395041481339931296117, −8.355932027401936860418397008017, −8.143713845836800345753243929629, −7.11292781221021161877521049935, −5.42361644357714142751149673014, −4.91211818234088658608788567179, −3.32352267429259266753661136233, −1.83861821219093617622060524495, 0.27486408724878850031783715106, 2.57607575502476320320990869220, 3.28377929136558672669822666972, 4.57296276432449334303438389448, 5.78106703785231550328921743602, 7.18160258534291627158032863873, 8.075912125518880473504005165760, 9.557627030906432368721605464398, 10.29005053811303552865421068263, 11.11815259861696543336538593091

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