Properties

Label 2-230-115.19-c2-0-12
Degree $2$
Conductor $230$
Sign $0.912 + 0.409i$
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  + (1.28 − 0.587i)2-s + (0.00972 + 0.0331i)3-s + (1.30 − 1.51i)4-s + (1.88 + 4.62i)5-s + (0.0319 + 0.0368i)6-s + (−0.869 − 6.04i)7-s + (0.796 − 2.71i)8-s + (7.57 − 4.86i)9-s + (5.15 + 4.84i)10-s + (−0.789 − 0.360i)11-s + (0.0627 + 0.0286i)12-s + (12.0 + 1.73i)13-s + (−4.67 − 7.26i)14-s + (−0.134 + 0.107i)15-s + (−0.569 − 3.95i)16-s + (19.6 + 22.6i)17-s + ⋯
L(s)  = 1  + (0.643 − 0.293i)2-s + (0.00324 + 0.0110i)3-s + (0.327 − 0.377i)4-s + (0.377 + 0.925i)5-s + (0.00532 + 0.00614i)6-s + (−0.124 − 0.864i)7-s + (0.0996 − 0.339i)8-s + (0.841 − 0.540i)9-s + (0.515 + 0.484i)10-s + (−0.0717 − 0.0327i)11-s + (0.00523 + 0.00238i)12-s + (0.928 + 0.133i)13-s + (−0.333 − 0.519i)14-s + (−0.00899 + 0.00717i)15-s + (−0.0355 − 0.247i)16-s + (1.15 + 1.33i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.912 + 0.409i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.912 + 0.409i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(230\)    =    \(2 \cdot 5 \cdot 23\)
Sign: $0.912 + 0.409i$
Analytic conductor: \(6.26704\)
Root analytic conductor: \(2.50340\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{230} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 230,\ (\ :1),\ 0.912 + 0.409i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.46938 - 0.528693i\)
\(L(\frac12)\) \(\approx\) \(2.46938 - 0.528693i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.28 + 0.587i)T \)
5 \( 1 + (-1.88 - 4.62i)T \)
23 \( 1 + (-20.5 + 10.2i)T \)
good3 \( 1 + (-0.00972 - 0.0331i)T + (-7.57 + 4.86i)T^{2} \)
7 \( 1 + (0.869 + 6.04i)T + (-47.0 + 13.8i)T^{2} \)
11 \( 1 + (0.789 + 0.360i)T + (79.2 + 91.4i)T^{2} \)
13 \( 1 + (-12.0 - 1.73i)T + (162. + 47.6i)T^{2} \)
17 \( 1 + (-19.6 - 22.6i)T + (-41.1 + 286. i)T^{2} \)
19 \( 1 + (7.05 + 6.11i)T + (51.3 + 357. i)T^{2} \)
29 \( 1 + (13.1 + 15.2i)T + (-119. + 832. i)T^{2} \)
31 \( 1 + (43.4 + 12.7i)T + (808. + 519. i)T^{2} \)
37 \( 1 + (-11.9 + 7.67i)T + (568. - 1.24e3i)T^{2} \)
41 \( 1 + (45.5 + 29.2i)T + (698. + 1.52e3i)T^{2} \)
43 \( 1 + (61.6 - 18.1i)T + (1.55e3 - 9.99e2i)T^{2} \)
47 \( 1 - 58.7iT - 2.20e3T^{2} \)
53 \( 1 + (-0.467 - 3.24i)T + (-2.69e3 + 791. i)T^{2} \)
59 \( 1 + (15.7 - 109. i)T + (-3.33e3 - 980. i)T^{2} \)
61 \( 1 + (26.5 - 90.4i)T + (-3.13e3 - 2.01e3i)T^{2} \)
67 \( 1 + (16.7 + 36.7i)T + (-2.93e3 + 3.39e3i)T^{2} \)
71 \( 1 + (31.2 + 68.3i)T + (-3.30e3 + 3.80e3i)T^{2} \)
73 \( 1 + (66.0 + 57.2i)T + (758. + 5.27e3i)T^{2} \)
79 \( 1 + (-66.1 - 9.51i)T + (5.98e3 + 1.75e3i)T^{2} \)
83 \( 1 + (-25.0 + 16.0i)T + (2.86e3 - 6.26e3i)T^{2} \)
89 \( 1 + (-4.94 - 16.8i)T + (-6.66e3 + 4.28e3i)T^{2} \)
97 \( 1 + (-112. - 72.6i)T + (3.90e3 + 8.55e3i)T^{2} \)
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

−11.96280190809241072488701258386, −10.65833314634094953097642876859, −10.48189958325741156234314151667, −9.253805590769616227780860876653, −7.61165644429163653679347091846, −6.68752490395415955301427230640, −5.83294450479988042720463466613, −4.12719389670726315502426109633, −3.33653169302335144845141583032, −1.50301623450878901924917078244, 1.67980024302968446977249751789, 3.42095588423311424497780025358, 5.00183683326652805226979228298, 5.50871723361775173316215134485, 6.88306326247840676792415955440, 8.061036765217728834943888118556, 9.037154975003208318318498203382, 10.01902247831192393742551135214, 11.34728075596657782221041208784, 12.31184432300720384636681026925

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