↑

Properties

Label 2-460-115.107-c1-0-3
Degree $2$
Conductor $460$
Sign $0.0753 - 0.997i$
Analytic cond. $3.67311$
Root an. cond. $1.91653$
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  + (1.80 + 0.674i)3-s + (0.366 + 2.20i)5-s + (−4.18 + 0.910i)7-s + (0.552 + 0.478i)9-s + (3.88 − 0.558i)11-s + (−1.43 + 6.57i)13-s + (−0.826 + 4.23i)15-s + (0.747 − 1.36i)17-s + (5.32 + 1.56i)19-s + (−8.18 − 1.17i)21-s + (1.70 + 4.48i)23-s + (−4.73 + 1.61i)25-s + (−2.10 − 3.84i)27-s + (−1.01 − 3.46i)29-s + (1.30 − 2.86i)31-s + ⋯
L(s)  = 1  + (1.04 + 0.389i)3-s + (0.163 + 0.986i)5-s + (−1.58 + 0.344i)7-s + (0.184 + 0.159i)9-s + (1.17 − 0.168i)11-s + (−0.396 + 1.82i)13-s + (−0.213 + 1.09i)15-s + (0.181 − 0.332i)17-s + (1.22 + 0.358i)19-s + (−1.78 − 0.256i)21-s + (0.354 + 0.934i)23-s + (−0.946 + 0.323i)25-s + (−0.404 − 0.740i)27-s + (−0.188 − 0.642i)29-s + (0.235 − 0.515i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0753 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0753 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(460\)    =    \(2^{2} \cdot 5 \cdot 23\)
Sign: $0.0753 - 0.997i$
Analytic conductor: \(3.67311\)
Root analytic conductor: \(1.91653\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{460} (337, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 460,\ (\ :1/2),\ 0.0753 - 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.25192 + 1.16092i\)
\(L(\frac12)\) \(\approx\) \(1.25192 + 1.16092i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (-0.366 - 2.20i)T \)
23 \( 1 + (-1.70 - 4.48i)T \)
good3 \( 1 + (-1.80 - 0.674i)T + (2.26 + 1.96i)T^{2} \)
7 \( 1 + (4.18 - 0.910i)T + (6.36 - 2.90i)T^{2} \)
11 \( 1 + (-3.88 + 0.558i)T + (10.5 - 3.09i)T^{2} \)
13 \( 1 + (1.43 - 6.57i)T + (-11.8 - 5.40i)T^{2} \)
17 \( 1 + (-0.747 + 1.36i)T + (-9.19 - 14.3i)T^{2} \)
19 \( 1 + (-5.32 - 1.56i)T + (15.9 + 10.2i)T^{2} \)
29 \( 1 + (1.01 + 3.46i)T + (-24.3 + 15.6i)T^{2} \)
31 \( 1 + (-1.30 + 2.86i)T + (-20.3 - 23.4i)T^{2} \)
37 \( 1 + (0.604 + 8.44i)T + (-36.6 + 5.26i)T^{2} \)
41 \( 1 + (-3.46 - 3.99i)T + (-5.83 + 40.5i)T^{2} \)
43 \( 1 + (-0.956 + 2.56i)T + (-32.4 - 28.1i)T^{2} \)
47 \( 1 + (-1.06 - 1.06i)T + 47iT^{2} \)
53 \( 1 + (1.93 + 8.89i)T + (-48.2 + 22.0i)T^{2} \)
59 \( 1 + (1.25 + 1.94i)T + (-24.5 + 53.6i)T^{2} \)
61 \( 1 + (-2.71 - 1.23i)T + (39.9 + 46.1i)T^{2} \)
67 \( 1 + (-4.79 - 6.40i)T + (-18.8 + 64.2i)T^{2} \)
71 \( 1 + (0.879 - 6.11i)T + (-68.1 - 20.0i)T^{2} \)
73 \( 1 + (1.67 - 0.912i)T + (39.4 - 61.4i)T^{2} \)
79 \( 1 + (-5.25 + 3.37i)T + (32.8 - 71.8i)T^{2} \)
83 \( 1 + (-16.7 + 1.19i)T + (82.1 - 11.8i)T^{2} \)
89 \( 1 + (3.58 + 7.85i)T + (-58.2 + 67.2i)T^{2} \)
97 \( 1 + (11.3 + 0.809i)T + (96.0 + 13.8i)T^{2} \)
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

−11.44489568814240984267000887685, −9.813779618012893107971914679558, −9.535137643502390060723734995977, −9.013731575337331906603928866580, −7.47345396648075848005057553193, −6.69840263878586588895182685243, −5.89403266040933057253879260349, −3.92927297075627084114525047336, −3.36024941081017618010106043371, −2.28559167323963975065099321173, 0.991976861820320061731583094372, 2.83558262604534172868187605659, 3.59399789154161587772089216225, 5.10162151635988826417370412227, 6.25859182179335254982930872273, 7.30933043917535208885213677703, 8.201030924000340152104238810790, 9.117699315016713648100324869782, 9.639937550483552266662723612046, 10.58508285552482179079767644238

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