Properties

Label 2-966-23.4-c1-0-16
Degree $2$
Conductor $966$
Sign $-0.904 + 0.427i$
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  + (−0.415 − 0.909i)2-s + (−0.959 + 0.281i)3-s + (−0.654 + 0.755i)4-s + (−0.512 − 0.329i)5-s + (0.654 + 0.755i)6-s + (−0.142 − 0.989i)7-s + (0.959 + 0.281i)8-s + (0.841 − 0.540i)9-s + (−0.0867 + 0.603i)10-s + (0.0366 − 0.0801i)11-s + (0.415 − 0.909i)12-s + (−0.291 + 2.02i)13-s + (−0.841 + 0.540i)14-s + (0.584 + 0.171i)15-s + (−0.142 − 0.989i)16-s + (0.448 + 0.517i)17-s + ⋯
L(s)  = 1  + (−0.293 − 0.643i)2-s + (−0.553 + 0.162i)3-s + (−0.327 + 0.377i)4-s + (−0.229 − 0.147i)5-s + (0.267 + 0.308i)6-s + (−0.0537 − 0.374i)7-s + (0.339 + 0.0996i)8-s + (0.280 − 0.180i)9-s + (−0.0274 + 0.190i)10-s + (0.0110 − 0.0241i)11-s + (0.119 − 0.262i)12-s + (−0.0808 + 0.562i)13-s + (−0.224 + 0.144i)14-s + (0.150 + 0.0443i)15-s + (−0.0355 − 0.247i)16-s + (0.108 + 0.125i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.904 + 0.427i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.904 + 0.427i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(966\)    =    \(2 \cdot 3 \cdot 7 \cdot 23\)
Sign: $-0.904 + 0.427i$
Analytic conductor: \(7.71354\)
Root analytic conductor: \(2.77732\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{966} (211, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 966,\ (\ :1/2),\ -0.904 + 0.427i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.123331 - 0.549355i\)
\(L(\frac12)\) \(\approx\) \(0.123331 - 0.549355i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.415 + 0.909i)T \)
3 \( 1 + (0.959 - 0.281i)T \)
7 \( 1 + (0.142 + 0.989i)T \)
23 \( 1 + (3.50 - 3.27i)T \)
good5 \( 1 + (0.512 + 0.329i)T + (2.07 + 4.54i)T^{2} \)
11 \( 1 + (-0.0366 + 0.0801i)T + (-7.20 - 8.31i)T^{2} \)
13 \( 1 + (0.291 - 2.02i)T + (-12.4 - 3.66i)T^{2} \)
17 \( 1 + (-0.448 - 0.517i)T + (-2.41 + 16.8i)T^{2} \)
19 \( 1 + (-4.55 + 5.26i)T + (-2.70 - 18.8i)T^{2} \)
29 \( 1 + (-0.230 - 0.265i)T + (-4.12 + 28.7i)T^{2} \)
31 \( 1 + (5.95 + 1.74i)T + (26.0 + 16.7i)T^{2} \)
37 \( 1 + (-7.13 + 4.58i)T + (15.3 - 33.6i)T^{2} \)
41 \( 1 + (3.96 + 2.54i)T + (17.0 + 37.2i)T^{2} \)
43 \( 1 + (1.36 - 0.400i)T + (36.1 - 23.2i)T^{2} \)
47 \( 1 + 6.29T + 47T^{2} \)
53 \( 1 + (1.03 + 7.17i)T + (-50.8 + 14.9i)T^{2} \)
59 \( 1 + (-0.496 + 3.45i)T + (-56.6 - 16.6i)T^{2} \)
61 \( 1 + (11.1 + 3.28i)T + (51.3 + 32.9i)T^{2} \)
67 \( 1 + (2.97 + 6.51i)T + (-43.8 + 50.6i)T^{2} \)
71 \( 1 + (5.66 + 12.4i)T + (-46.4 + 53.6i)T^{2} \)
73 \( 1 + (-0.0217 + 0.0251i)T + (-10.3 - 72.2i)T^{2} \)
79 \( 1 + (-1.68 + 11.7i)T + (-75.7 - 22.2i)T^{2} \)
83 \( 1 + (12.0 - 7.74i)T + (34.4 - 75.4i)T^{2} \)
89 \( 1 + (5.61 - 1.64i)T + (74.8 - 48.1i)T^{2} \)
97 \( 1 + (-10.8 - 6.99i)T + (40.2 + 88.2i)T^{2} \)
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

−9.659559554182833074900171074281, −9.167905526534105162311328067567, −7.975861036597905498780626619240, −7.26261751565342466226905419968, −6.21986916454323334975567156956, −5.09383114937063691910175743716, −4.24223738393496693387711577717, −3.28406205716277659572634212386, −1.81724973309312619233118224303, −0.32645905527197898126913246221, 1.43954668933624380715257303040, 3.13012845360625262660901069750, 4.38891638820892912207846641599, 5.52796613978898116011316461586, 5.97446230141947248612309559012, 7.09195190803595172281545363940, 7.76629515487875669122406790155, 8.522777816799905050481790616480, 9.656501834416934451933900175926, 10.16452120939082458689473140470

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