Properties

Label 2-230-115.63-c1-0-8
Degree $2$
Conductor $230$
Sign $0.933 + 0.358i$
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  + (0.349 + 0.936i)2-s + (−0.926 − 1.69i)3-s + (−0.755 + 0.654i)4-s + (2.12 − 0.681i)5-s + (1.26 − 1.46i)6-s + (−0.983 + 0.735i)7-s + (−0.877 − 0.479i)8-s + (−0.400 + 0.623i)9-s + (1.38 + 1.75i)10-s + (3.08 − 1.40i)11-s + (1.81 + 0.675i)12-s + (4.09 − 5.47i)13-s + (−1.03 − 0.663i)14-s + (−3.13 − 2.98i)15-s + (0.142 − 0.989i)16-s + (−3.00 + 0.214i)17-s + ⋯
L(s)  = 1  + (0.247 + 0.662i)2-s + (−0.535 − 0.980i)3-s + (−0.377 + 0.327i)4-s + (0.952 − 0.304i)5-s + (0.517 − 0.596i)6-s + (−0.371 + 0.278i)7-s + (−0.310 − 0.169i)8-s + (−0.133 + 0.207i)9-s + (0.437 + 0.555i)10-s + (0.929 − 0.424i)11-s + (0.523 + 0.195i)12-s + (1.13 − 1.51i)13-s + (−0.276 − 0.177i)14-s + (−0.808 − 0.770i)15-s + (0.0355 − 0.247i)16-s + (−0.728 + 0.0520i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.933 + 0.358i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.933 + 0.358i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(230\)    =    \(2 \cdot 5 \cdot 23\)
Sign: $0.933 + 0.358i$
Analytic conductor: \(1.83655\)
Root analytic conductor: \(1.35519\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{230} (63, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 230,\ (\ :1/2),\ 0.933 + 0.358i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.29537 - 0.239885i\)
\(L(\frac12)\) \(\approx\) \(1.29537 - 0.239885i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.349 - 0.936i)T \)
5 \( 1 + (-2.12 + 0.681i)T \)
23 \( 1 + (2.74 - 3.93i)T \)
good3 \( 1 + (0.926 + 1.69i)T + (-1.62 + 2.52i)T^{2} \)
7 \( 1 + (0.983 - 0.735i)T + (1.97 - 6.71i)T^{2} \)
11 \( 1 + (-3.08 + 1.40i)T + (7.20 - 8.31i)T^{2} \)
13 \( 1 + (-4.09 + 5.47i)T + (-3.66 - 12.4i)T^{2} \)
17 \( 1 + (3.00 - 0.214i)T + (16.8 - 2.41i)T^{2} \)
19 \( 1 + (-1.88 - 2.17i)T + (-2.70 + 18.8i)T^{2} \)
29 \( 1 + (-3.99 - 3.46i)T + (4.12 + 28.7i)T^{2} \)
31 \( 1 + (5.92 - 1.74i)T + (26.0 - 16.7i)T^{2} \)
37 \( 1 + (-1.42 - 6.54i)T + (-33.6 + 15.3i)T^{2} \)
41 \( 1 + (2.34 - 1.50i)T + (17.0 - 37.2i)T^{2} \)
43 \( 1 + (7.40 - 4.04i)T + (23.2 - 36.1i)T^{2} \)
47 \( 1 + (-5.79 + 5.79i)T - 47iT^{2} \)
53 \( 1 + (6.30 + 8.41i)T + (-14.9 + 50.8i)T^{2} \)
59 \( 1 + (-7.66 + 1.10i)T + (56.6 - 16.6i)T^{2} \)
61 \( 1 + (-0.704 - 2.39i)T + (-51.3 + 32.9i)T^{2} \)
67 \( 1 + (8.58 - 3.20i)T + (50.6 - 43.8i)T^{2} \)
71 \( 1 + (1.74 - 3.83i)T + (-46.4 - 53.6i)T^{2} \)
73 \( 1 + (0.0550 - 0.769i)T + (-72.2 - 10.3i)T^{2} \)
79 \( 1 + (-0.551 - 3.83i)T + (-75.7 + 22.2i)T^{2} \)
83 \( 1 + (-11.8 + 2.57i)T + (75.4 - 34.4i)T^{2} \)
89 \( 1 + (-10.7 - 3.16i)T + (74.8 + 48.1i)T^{2} \)
97 \( 1 + (6.29 + 1.36i)T + (88.2 + 40.2i)T^{2} \)
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

−12.44923624690601520882191488005, −11.46493153446318170074605452350, −10.13166450337686367414465778187, −9.014712777633444524520635572185, −8.093859861930972950101159717108, −6.71556261250986611829975854780, −6.11470096551673583392228244626, −5.36653851658200513222801251496, −3.44896330486716922625059689851, −1.30897084005250089839376757854, 1.95448895507463886964328239334, 3.82085455683234611356819752717, 4.62316685313673351789179784906, 5.99881825685927180107956212684, 6.81297644124284182995884182958, 9.030097714586642025282888112203, 9.484633548035516556226018422557, 10.48466045334718170600660770887, 11.11754786475975369579822991323, 12.00593479542226122881698435297

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