↑

Properties

Label 2-966-21.20-c1-0-8
Degree $2$
Conductor $966$
Sign $-0.0625 - 0.998i$
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  + i·2-s + (−0.375 − 1.69i)3-s − 4-s − 0.513·5-s + (1.69 − 0.375i)6-s + (−2.61 − 0.410i)7-s − i·8-s + (−2.71 + 1.26i)9-s − 0.513i·10-s − 3.12i·11-s + (0.375 + 1.69i)12-s + 5.94i·13-s + (0.410 − 2.61i)14-s + (0.192 + 0.868i)15-s + 16-s + 5.76·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (−0.216 − 0.976i)3-s − 0.5·4-s − 0.229·5-s + (0.690 − 0.153i)6-s + (−0.987 − 0.155i)7-s − 0.353i·8-s + (−0.906 + 0.423i)9-s − 0.162i·10-s − 0.942i·11-s + (0.108 + 0.488i)12-s + 1.64i·13-s + (0.109 − 0.698i)14-s + (0.0497 + 0.224i)15-s + 0.250·16-s + 1.39·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0625 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0625 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(966\)    =    \(2 \cdot 3 \cdot 7 \cdot 23\)
Sign: $-0.0625 - 0.998i$
Analytic conductor: \(7.71354\)
Root analytic conductor: \(2.77732\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{966} (461, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 966,\ (\ :1/2),\ -0.0625 - 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.517417 + 0.550881i\)
\(L(\frac12)\) \(\approx\) \(0.517417 + 0.550881i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 + (0.375 + 1.69i)T \)
7 \( 1 + (2.61 + 0.410i)T \)
23 \( 1 + iT \)
good5 \( 1 + 0.513T + 5T^{2} \)
11 \( 1 + 3.12iT - 11T^{2} \)
13 \( 1 - 5.94iT - 13T^{2} \)
17 \( 1 - 5.76T + 17T^{2} \)
19 \( 1 + 0.598iT - 19T^{2} \)
29 \( 1 - 7.46iT - 29T^{2} \)
31 \( 1 - 7.72iT - 31T^{2} \)
37 \( 1 + 1.50T + 37T^{2} \)
41 \( 1 + 1.20T + 41T^{2} \)
43 \( 1 - 5.20T + 43T^{2} \)
47 \( 1 - 6.27T + 47T^{2} \)
53 \( 1 - 11.4iT - 53T^{2} \)
59 \( 1 - 11.2T + 59T^{2} \)
61 \( 1 + 0.414iT - 61T^{2} \)
67 \( 1 + 6.23T + 67T^{2} \)
71 \( 1 - 4.92iT - 71T^{2} \)
73 \( 1 - 3.88iT - 73T^{2} \)
79 \( 1 + 12.9T + 79T^{2} \)
83 \( 1 + 11.5T + 83T^{2} \)
89 \( 1 - 3.08T + 89T^{2} \)
97 \( 1 - 4.96iT - 97T^{2} \)
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

−10.15453276379883983720073138947, −9.068439308274712108241867048918, −8.549298039930711665049961510981, −7.40400829972999615140016636820, −6.93975909824209742846346209478, −6.09796933764666502387875168764, −5.40746486671177514725881483938, −3.97793472199494112133133645036, −2.95087964478376573147354662170, −1.21808720220877414564996283889, 0.40685644956083262934561404999, 2.54625077405553448842916852153, 3.48025948661607530400155815433, 4.18579001373028151307214323609, 5.46712968049286026204793212669, 5.92624373705793233372391736158, 7.49120313919246866730949221026, 8.260008282307823976272663876030, 9.438818216043232967726757826543, 10.02519711208684812227381732987

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