Properties

Label 2-966-21.20-c1-0-4
Degree $2$
Conductor $966$
Sign $-0.654 - 0.755i$
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 + 1.73i·3-s − 4-s − 1.73·5-s + 1.73·6-s + (2 − 1.73i)7-s + i·8-s − 2.99·9-s + 1.73i·10-s − 1.73i·12-s + 1.73i·13-s + (−1.73 − 2i)14-s − 2.99i·15-s + 16-s − 3.46·17-s + 2.99i·18-s + ⋯
L(s)  = 1  − 0.707i·2-s + 0.999i·3-s − 0.5·4-s − 0.774·5-s + 0.707·6-s + (0.755 − 0.654i)7-s + 0.353i·8-s − 0.999·9-s + 0.547i·10-s − 0.499i·12-s + 0.480i·13-s + (−0.462 − 0.534i)14-s − 0.774i·15-s + 0.250·16-s − 0.840·17-s + 0.707i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.654 - 0.755i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.654 - 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(966\)    =    \(2 \cdot 3 \cdot 7 \cdot 23\)
Sign: $-0.654 - 0.755i$
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: \((2,\ 966,\ (\ :1/2),\ -0.654 - 0.755i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.203029 + 0.444411i\)
\(L(\frac12)\) \(\approx\) \(0.203029 + 0.444411i\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.73iT \)
7 \( 1 + (-2 + 1.73i)T \)
23 \( 1 - iT \)
good5 \( 1 + 1.73T + 5T^{2} \)
11 \( 1 - 11T^{2} \)
13 \( 1 - 1.73iT - 13T^{2} \)
17 \( 1 + 3.46T + 17T^{2} \)
19 \( 1 - 6.92iT - 19T^{2} \)
29 \( 1 + 9iT - 29T^{2} \)
31 \( 1 - 10.3iT - 31T^{2} \)
37 \( 1 + 5T + 37T^{2} \)
41 \( 1 + 12.1T + 41T^{2} \)
43 \( 1 - T + 43T^{2} \)
47 \( 1 + 8.66T + 47T^{2} \)
53 \( 1 - 12iT - 53T^{2} \)
59 \( 1 - 3.46T + 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 + 4T + 67T^{2} \)
71 \( 1 - 12iT - 71T^{2} \)
73 \( 1 - 6.92iT - 73T^{2} \)
79 \( 1 + 10T + 79T^{2} \)
83 \( 1 + 3.46T + 83T^{2} \)
89 \( 1 - 3.46T + 89T^{2} \)
97 \( 1 + 12.1iT - 97T^{2} \)
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

−10.29830337512985814919375355078, −9.824722748357661531068942393906, −8.565405649765283100892921738951, −8.238889663770030904643678950823, −7.07850844113640493192684156298, −5.72506712664060025412618134921, −4.63962469859591386777720407666, −4.09279397665848444004050171896, −3.29809003246557298704333918087, −1.73782153158932564270454481239, 0.22201275981737702551831725562, 1.96982109007826850296924808870, 3.26446235748117919696487282760, 4.70410686785791610149054076297, 5.41047220927664224654783423445, 6.53404776537550041285097144429, 7.16949163931782490797876191436, 8.045722799724118625413908132491, 8.534914062152172133142293588109, 9.290365232294858221586791552003

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