Properties

Label 2-230-115.107-c1-0-11
Degree $2$
Conductor $230$
Sign $-0.895 + 0.445i$
Analytic cond. $1.83655$
Root an. cond. $1.35519$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.800 − 0.599i)2-s + (−1.19 − 0.447i)3-s + (0.281 − 0.959i)4-s + (−1.17 − 1.90i)5-s + (−1.22 + 0.360i)6-s + (−4.10 + 0.892i)7-s + (−0.349 − 0.936i)8-s + (−1.02 − 0.892i)9-s + (−2.08 − 0.818i)10-s + (2.30 − 0.330i)11-s + (−0.766 + 1.02i)12-s + (−0.202 + 0.932i)13-s + (−2.74 + 3.17i)14-s + (0.557 + 2.80i)15-s + (−0.841 − 0.540i)16-s + (3.53 − 6.47i)17-s + ⋯
L(s)  = 1  + (0.566 − 0.423i)2-s + (−0.692 − 0.258i)3-s + (0.140 − 0.479i)4-s + (−0.525 − 0.850i)5-s + (−0.501 + 0.147i)6-s + (−1.55 + 0.337i)7-s + (−0.123 − 0.331i)8-s + (−0.343 − 0.297i)9-s + (−0.657 − 0.258i)10-s + (0.694 − 0.0997i)11-s + (−0.221 + 0.295i)12-s + (−0.0562 + 0.258i)13-s + (−0.734 + 0.847i)14-s + (0.144 + 0.724i)15-s + (−0.210 − 0.135i)16-s + (0.857 − 1.56i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.895 + 0.445i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.895 + 0.445i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(230\)    =    \(2 \cdot 5 \cdot 23\)
Sign: $-0.895 + 0.445i$
Analytic conductor: \(1.83655\)
Root analytic conductor: \(1.35519\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{230} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 230,\ (\ :1/2),\ -0.895 + 0.445i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.184771 - 0.785795i\)
\(L(\frac12)\) \(\approx\) \(0.184771 - 0.785795i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.800 + 0.599i)T \)
5 \( 1 + (1.17 + 1.90i)T \)
23 \( 1 + (4.75 + 0.628i)T \)
good3 \( 1 + (1.19 + 0.447i)T + (2.26 + 1.96i)T^{2} \)
7 \( 1 + (4.10 - 0.892i)T + (6.36 - 2.90i)T^{2} \)
11 \( 1 + (-2.30 + 0.330i)T + (10.5 - 3.09i)T^{2} \)
13 \( 1 + (0.202 - 0.932i)T + (-11.8 - 5.40i)T^{2} \)
17 \( 1 + (-3.53 + 6.47i)T + (-9.19 - 14.3i)T^{2} \)
19 \( 1 + (-2.40 - 0.706i)T + (15.9 + 10.2i)T^{2} \)
29 \( 1 + (1.30 + 4.45i)T + (-24.3 + 15.6i)T^{2} \)
31 \( 1 + (-4.15 + 9.09i)T + (-20.3 - 23.4i)T^{2} \)
37 \( 1 + (0.541 + 7.57i)T + (-36.6 + 5.26i)T^{2} \)
41 \( 1 + (-5.61 - 6.47i)T + (-5.83 + 40.5i)T^{2} \)
43 \( 1 + (2.77 - 7.43i)T + (-32.4 - 28.1i)T^{2} \)
47 \( 1 + (-7.75 - 7.75i)T + 47iT^{2} \)
53 \( 1 + (0.963 + 4.42i)T + (-48.2 + 22.0i)T^{2} \)
59 \( 1 + (-2.26 - 3.53i)T + (-24.5 + 53.6i)T^{2} \)
61 \( 1 + (6.04 + 2.76i)T + (39.9 + 46.1i)T^{2} \)
67 \( 1 + (6.85 + 9.15i)T + (-18.8 + 64.2i)T^{2} \)
71 \( 1 + (-0.111 + 0.774i)T + (-68.1 - 20.0i)T^{2} \)
73 \( 1 + (-2.62 + 1.43i)T + (39.4 - 61.4i)T^{2} \)
79 \( 1 + (-0.534 + 0.343i)T + (32.8 - 71.8i)T^{2} \)
83 \( 1 + (9.40 - 0.672i)T + (82.1 - 11.8i)T^{2} \)
89 \( 1 + (-1.06 - 2.34i)T + (-58.2 + 67.2i)T^{2} \)
97 \( 1 + (0.460 + 0.0329i)T + (96.0 + 13.8i)T^{2} \)
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

−11.94664409118027505877320638874, −11.37343905874073246803575751543, −9.609507634161236504289673425362, −9.358451761389842307018999456911, −7.63001283794656012608118959949, −6.29781391476563974059482905687, −5.67157345773241428907601346546, −4.23865796289510395256874135390, −3.02038787859571899858105775647, −0.59613864139515593806353628792, 3.14811512949486847703030957772, 3.98044049032814238358870419279, 5.63449111450769324217743445423, 6.42325516750224321618630809322, 7.25404145640399869500999972744, 8.535029804427184237399209649653, 10.13591331515317927385213127985, 10.58384700345536261015967809639, 11.93261329551910424784529588101, 12.38593598854635099916153907999

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