↑

Properties

Label 2-966-161.19-c1-0-29
Degree $2$
Conductor $966$
Sign $0.388 + 0.921i$
Analytic cond. $7.71354$
Root an. cond. $2.77732$
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  + (0.995 + 0.0950i)2-s + (0.690 − 0.723i)3-s + (0.981 + 0.189i)4-s + (1.83 − 0.943i)5-s + (0.755 − 0.654i)6-s + (−2.58 − 0.565i)7-s + (0.959 + 0.281i)8-s + (−0.0475 − 0.998i)9-s + (1.91 − 0.765i)10-s + (−0.336 − 3.52i)11-s + (0.814 − 0.580i)12-s + (−2.03 − 0.292i)13-s + (−2.51 − 0.808i)14-s + (0.580 − 1.97i)15-s + (0.928 + 0.371i)16-s + (1.97 − 5.71i)17-s + ⋯
L(s)  = 1  + (0.703 + 0.0672i)2-s + (0.398 − 0.417i)3-s + (0.490 + 0.0946i)4-s + (0.818 − 0.422i)5-s + (0.308 − 0.267i)6-s + (−0.976 − 0.213i)7-s + (0.339 + 0.0996i)8-s + (−0.0158 − 0.332i)9-s + (0.604 − 0.242i)10-s + (−0.101 − 1.06i)11-s + (0.235 − 0.167i)12-s + (−0.564 − 0.0811i)13-s + (−0.673 − 0.216i)14-s + (0.149 − 0.510i)15-s + (0.232 + 0.0929i)16-s + (0.479 − 1.38i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.388 + 0.921i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.388 + 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(966\)    =    \(2 \cdot 3 \cdot 7 \cdot 23\)
Sign: $0.388 + 0.921i$
Analytic conductor: \(7.71354\)
Root analytic conductor: \(2.77732\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{966} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 966,\ (\ :1/2),\ 0.388 + 0.921i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.34979 - 1.55878i\)
\(L(\frac12)\) \(\approx\) \(2.34979 - 1.55878i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.995 - 0.0950i)T \)
3 \( 1 + (-0.690 + 0.723i)T \)
7 \( 1 + (2.58 + 0.565i)T \)
23 \( 1 + (-3.99 + 2.64i)T \)
good5 \( 1 + (-1.83 + 0.943i)T + (2.90 - 4.07i)T^{2} \)
11 \( 1 + (0.336 + 3.52i)T + (-10.8 + 2.08i)T^{2} \)
13 \( 1 + (2.03 + 0.292i)T + (12.4 + 3.66i)T^{2} \)
17 \( 1 + (-1.97 + 5.71i)T + (-13.3 - 10.5i)T^{2} \)
19 \( 1 + (-0.991 - 2.86i)T + (-14.9 + 11.7i)T^{2} \)
29 \( 1 + (-0.0135 - 0.0156i)T + (-4.12 + 28.7i)T^{2} \)
31 \( 1 + (-7.10 + 1.72i)T + (27.5 - 14.2i)T^{2} \)
37 \( 1 + (0.158 - 0.00756i)T + (36.8 - 3.51i)T^{2} \)
41 \( 1 + (6.38 - 9.94i)T + (-17.0 - 37.2i)T^{2} \)
43 \( 1 + (-0.590 - 2.01i)T + (-36.1 + 23.2i)T^{2} \)
47 \( 1 + (-6.67 - 3.85i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.81 - 3.57i)T + (-12.4 + 51.5i)T^{2} \)
59 \( 1 + (1.60 + 4.01i)T + (-42.7 + 40.7i)T^{2} \)
61 \( 1 + (9.65 - 9.20i)T + (2.90 - 60.9i)T^{2} \)
67 \( 1 + (0.752 + 0.535i)T + (21.9 + 63.3i)T^{2} \)
71 \( 1 + (2.49 + 5.46i)T + (-46.4 + 53.6i)T^{2} \)
73 \( 1 + (1.10 - 5.72i)T + (-67.7 - 27.1i)T^{2} \)
79 \( 1 + (-4.70 + 5.98i)T + (-18.6 - 76.7i)T^{2} \)
83 \( 1 + (-4.70 + 3.02i)T + (34.4 - 75.4i)T^{2} \)
89 \( 1 + (1.84 - 7.62i)T + (-79.1 - 40.7i)T^{2} \)
97 \( 1 + (-4.96 - 3.19i)T + (40.2 + 88.2i)T^{2} \)
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

−9.740447303814238697919339830955, −9.175176823670497988859207705239, −8.078754260439230440985468073523, −7.18941041082144851564422099221, −6.30307704950755435275491609879, −5.63988669717268756555894968300, −4.64232702212117252440199666432, −3.23182563420994046510624113453, −2.68116002693334437702393995525, −1.02763808999452518898610474964, 1.98290670911032363901709897188, 2.85308533322612760813863453097, 3.81599916779016798127619924572, 4.91944593740798559809734588243, 5.79390599688215948236172716828, 6.68762946345159304464031319327, 7.37835951851232421955446419369, 8.675955950500768500873649934455, 9.621929403339877911651639482780, 10.13284509953081215269829157135

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