Properties

Label 2-966-161.160-c1-0-9
Degree $2$
Conductor $966$
Sign $0.551 + 0.834i$
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  − 2-s + i·3-s + 4-s − 2.64·5-s i·6-s − 2.64·7-s − 8-s − 9-s + 2.64·10-s + i·12-s + 5i·13-s + 2.64·14-s − 2.64i·15-s + 16-s − 5.29·17-s + 18-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577i·3-s + 0.5·4-s − 1.18·5-s − 0.408i·6-s − 0.999·7-s − 0.353·8-s − 0.333·9-s + 0.836·10-s + 0.288i·12-s + 1.38i·13-s + 0.707·14-s − 0.683i·15-s + 0.250·16-s − 1.28·17-s + 0.235·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.551 + 0.834i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.551 + 0.834i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(966\)    =    \(2 \cdot 3 \cdot 7 \cdot 23\)
Sign: $0.551 + 0.834i$
Analytic conductor: \(7.71354\)
Root analytic conductor: \(2.77732\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{966} (643, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 966,\ (\ :1/2),\ 0.551 + 0.834i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.357178 - 0.191990i\)
\(L(\frac12)\) \(\approx\) \(0.357178 - 0.191990i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 - iT \)
7 \( 1 + 2.64T \)
23 \( 1 + (-4 + 2.64i)T \)
good5 \( 1 + 2.64T + 5T^{2} \)
11 \( 1 - 11T^{2} \)
13 \( 1 - 5iT - 13T^{2} \)
17 \( 1 + 5.29T + 17T^{2} \)
19 \( 1 - 5.29T + 19T^{2} \)
29 \( 1 + T + 29T^{2} \)
31 \( 1 + 4iT - 31T^{2} \)
37 \( 1 + 7.93iT - 37T^{2} \)
41 \( 1 + 9iT - 41T^{2} \)
43 \( 1 - 2.64iT - 43T^{2} \)
47 \( 1 + 13iT - 47T^{2} \)
53 \( 1 + 5.29iT - 53T^{2} \)
59 \( 1 - 14iT - 59T^{2} \)
61 \( 1 - 10.5T + 61T^{2} \)
67 \( 1 - 5.29iT - 67T^{2} \)
71 \( 1 - 6T + 71T^{2} \)
73 \( 1 - 4iT - 73T^{2} \)
79 \( 1 + 5.29iT - 79T^{2} \)
83 \( 1 + 15.8T + 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 - 2.64T + 97T^{2} \)
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

−9.731983502952521712384197782467, −9.056142110904500014175391223673, −8.510993704844601107405540156944, −7.17872244833444248515363123187, −6.92029629010205902909642240010, −5.61990486367059694957639699130, −4.27833013905006295223709392656, −3.64612377465148979653296831469, −2.39724569418101838105614407726, −0.30390715905302764015405052767, 0.964104969292270723355804869343, 2.83983782108463324179387374733, 3.47146858289704751004614479195, 4.97578329272074571040944417996, 6.17733635976808512340893295661, 6.99684911924331932258014092444, 7.69062820170516949488930218902, 8.333825815848299020992563566284, 9.264477680213359865685090446031, 10.05054210252501538651604715293

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