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Properties

Label 2-966-7.2-c1-0-12
Degree $2$
Conductor $966$
Sign $0.867 - 0.497i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 + 0.866i)2-s + (−0.5 − 0.866i)3-s + (−0.499 − 0.866i)4-s + (1.91 − 3.31i)5-s + 0.999·6-s + (1.16 + 2.37i)7-s + 0.999·8-s + (−0.499 + 0.866i)9-s + (1.91 + 3.31i)10-s + (2.55 + 4.42i)11-s + (−0.499 + 0.866i)12-s + 0.0556·13-s + (−2.63 − 0.175i)14-s − 3.82·15-s + (−0.5 + 0.866i)16-s + (3.38 + 5.86i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.288 − 0.499i)3-s + (−0.249 − 0.433i)4-s + (0.856 − 1.48i)5-s + 0.408·6-s + (0.441 + 0.897i)7-s + 0.353·8-s + (−0.166 + 0.288i)9-s + (0.605 + 1.04i)10-s + (0.770 + 1.33i)11-s + (−0.144 + 0.249i)12-s + 0.0154·13-s + (−0.705 − 0.0469i)14-s − 0.988·15-s + (−0.125 + 0.216i)16-s + (0.821 + 1.42i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.867 - 0.497i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.867 - 0.497i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(966\)    =    \(2 \cdot 3 \cdot 7 \cdot 23\)
Sign: $0.867 - 0.497i$
Analytic conductor: \(7.71354\)
Root analytic conductor: \(2.77732\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{966} (415, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 966,\ (\ :1/2),\ 0.867 - 0.497i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.43438 + 0.381957i\)
\(L(\frac12)\) \(\approx\) \(1.43438 + 0.381957i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 - 0.866i)T \)
3 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + (-1.16 - 2.37i)T \)
23 \( 1 + (0.5 - 0.866i)T \)
good5 \( 1 + (-1.91 + 3.31i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-2.55 - 4.42i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 0.0556T + 13T^{2} \)
17 \( 1 + (-3.38 - 5.86i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.08 - 3.60i)T + (-9.5 - 16.4i)T^{2} \)
29 \( 1 - 2.94T + 29T^{2} \)
31 \( 1 + (-1.60 - 2.78i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-4.02 + 6.97i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 4.16T + 41T^{2} \)
43 \( 1 - 12.5T + 43T^{2} \)
47 \( 1 + (5.99 - 10.3i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (2.85 + 4.95i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (4.30 + 7.44i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.88 + 5.00i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.07 + 10.5i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 11.9T + 71T^{2} \)
73 \( 1 + (-1.10 - 1.92i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.47 + 2.54i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 14.7T + 83T^{2} \)
89 \( 1 + (1.19 - 2.07i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 4.60T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.684099914403277475220316449772, −9.263757946658774489864096169427, −8.302487914138845829957440297797, −7.85365412606210980576947702194, −6.44102783768252728653078007100, −5.88514270611825007166019218246, −5.09585744215620993976510650394, −4.25814966685418883852841273139, −1.94185073495420425109971288833, −1.38338998166535556293017113178, 0.943199552988592316438381048486, 2.63681210996530170840361930766, 3.35351805398601735874082083054, 4.45085166223788997054728970493, 5.69586091899213840655104329210, 6.60613566110512116683703391353, 7.31745149248780167798351834243, 8.472467225234776899331481075961, 9.469324485903229558975214846621, 10.09364355513415919603334866593

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