Properties

Label 2-966-23.12-c1-0-17
Degree $2$
Conductor $966$
Sign $-0.114 + 0.993i$
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  + (−0.841 + 0.540i)2-s + (0.142 − 0.989i)3-s + (0.415 − 0.909i)4-s + (2.36 − 0.694i)5-s + (0.415 + 0.909i)6-s + (−0.654 − 0.755i)7-s + (0.142 + 0.989i)8-s + (−0.959 − 0.281i)9-s + (−1.61 + 1.86i)10-s + (−2.22 − 1.43i)11-s + (−0.841 − 0.540i)12-s + (−0.672 + 0.776i)13-s + (0.959 + 0.281i)14-s + (−0.350 − 2.43i)15-s + (−0.654 − 0.755i)16-s + (−0.649 − 1.42i)17-s + ⋯
L(s)  = 1  + (−0.594 + 0.382i)2-s + (0.0821 − 0.571i)3-s + (0.207 − 0.454i)4-s + (1.05 − 0.310i)5-s + (0.169 + 0.371i)6-s + (−0.247 − 0.285i)7-s + (0.0503 + 0.349i)8-s + (−0.319 − 0.0939i)9-s + (−0.510 + 0.589i)10-s + (−0.671 − 0.431i)11-s + (−0.242 − 0.156i)12-s + (−0.186 + 0.215i)13-s + (0.256 + 0.0752i)14-s + (−0.0905 − 0.629i)15-s + (−0.163 − 0.188i)16-s + (−0.157 − 0.344i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.114 + 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.114 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(966\)    =    \(2 \cdot 3 \cdot 7 \cdot 23\)
Sign: $-0.114 + 0.993i$
Analytic conductor: \(7.71354\)
Root analytic conductor: \(2.77732\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{966} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 966,\ (\ :1/2),\ -0.114 + 0.993i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.759015 - 0.851859i\)
\(L(\frac12)\) \(\approx\) \(0.759015 - 0.851859i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.841 - 0.540i)T \)
3 \( 1 + (-0.142 + 0.989i)T \)
7 \( 1 + (0.654 + 0.755i)T \)
23 \( 1 + (-2.88 + 3.82i)T \)
good5 \( 1 + (-2.36 + 0.694i)T + (4.20 - 2.70i)T^{2} \)
11 \( 1 + (2.22 + 1.43i)T + (4.56 + 10.0i)T^{2} \)
13 \( 1 + (0.672 - 0.776i)T + (-1.85 - 12.8i)T^{2} \)
17 \( 1 + (0.649 + 1.42i)T + (-11.1 + 12.8i)T^{2} \)
19 \( 1 + (-2.76 + 6.05i)T + (-12.4 - 14.3i)T^{2} \)
29 \( 1 + (-2.11 - 4.62i)T + (-18.9 + 21.9i)T^{2} \)
31 \( 1 + (0.723 + 5.03i)T + (-29.7 + 8.73i)T^{2} \)
37 \( 1 + (9.74 + 2.86i)T + (31.1 + 20.0i)T^{2} \)
41 \( 1 + (5.32 - 1.56i)T + (34.4 - 22.1i)T^{2} \)
43 \( 1 + (0.397 - 2.76i)T + (-41.2 - 12.1i)T^{2} \)
47 \( 1 - 8.64T + 47T^{2} \)
53 \( 1 + (0.976 + 1.12i)T + (-7.54 + 52.4i)T^{2} \)
59 \( 1 + (0.174 - 0.201i)T + (-8.39 - 58.3i)T^{2} \)
61 \( 1 + (1.90 + 13.2i)T + (-58.5 + 17.1i)T^{2} \)
67 \( 1 + (0.344 - 0.221i)T + (27.8 - 60.9i)T^{2} \)
71 \( 1 + (-7.30 + 4.69i)T + (29.4 - 64.5i)T^{2} \)
73 \( 1 + (-1.03 + 2.26i)T + (-47.8 - 55.1i)T^{2} \)
79 \( 1 + (7.23 - 8.34i)T + (-11.2 - 78.1i)T^{2} \)
83 \( 1 + (-14.5 - 4.28i)T + (69.8 + 44.8i)T^{2} \)
89 \( 1 + (-1.22 + 8.54i)T + (-85.3 - 25.0i)T^{2} \)
97 \( 1 + (4.06 - 1.19i)T + (81.6 - 52.4i)T^{2} \)
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

−9.532610376999373307347315753785, −9.047188939779029398252994225569, −8.176455317939609088533590775464, −7.13897110631272898025324927460, −6.61986176834048187033934410023, −5.56550661286382019344081691887, −4.88954932611025089743280471997, −3.03517296711656921132623071617, −2.00190512483848022782037728449, −0.61906035603042443556578733331, 1.70745801223789330564027111599, 2.73446536406558917367457094523, 3.70907992714266008402920946090, 5.15758609709516178486958910310, 5.83953101887657030944332386344, 6.93595540550480194442195509561, 7.907939082490656973912343162408, 8.827157686966060386246755468952, 9.585823536788306397194306552215, 10.38454780479964561962304975388

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