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Properties

Label 2-966-7.4-c1-0-21
Degree $2$
Conductor $966$
Sign $0.0725 + 0.997i$
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 + (−0.207 − 0.358i)5-s − 0.999·6-s + (−2.62 + 0.358i)7-s − 0.999·8-s + (−0.499 − 0.866i)9-s + (0.207 − 0.358i)10-s + (−1 + 1.73i)11-s + (−0.499 − 0.866i)12-s − 1.82·13-s + (−1.62 − 2.09i)14-s + 0.414·15-s + (−0.5 − 0.866i)16-s + (1.62 − 2.80i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.288 + 0.499i)3-s + (−0.249 + 0.433i)4-s + (−0.0926 − 0.160i)5-s − 0.408·6-s + (−0.990 + 0.135i)7-s − 0.353·8-s + (−0.166 − 0.288i)9-s + (0.0654 − 0.113i)10-s + (−0.301 + 0.522i)11-s + (−0.144 − 0.249i)12-s − 0.507·13-s + (−0.433 − 0.558i)14-s + 0.106·15-s + (−0.125 − 0.216i)16-s + (0.393 − 0.681i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0725 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0725 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(966\)    =    \(2 \cdot 3 \cdot 7 \cdot 23\)
Sign: $0.0725 + 0.997i$
Analytic conductor: \(7.71354\)
Root analytic conductor: \(2.77732\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{966} (277, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 966,\ (\ :1/2),\ 0.0725 + 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.174909 - 0.162646i\)
\(L(\frac12)\) \(\approx\) \(0.174909 - 0.162646i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.62 - 0.358i)T \)
23 \( 1 + (0.5 + 0.866i)T \)
good5 \( 1 + (0.207 + 0.358i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (1 - 1.73i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 1.82T + 13T^{2} \)
17 \( 1 + (-1.62 + 2.80i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.82 + 3.16i)T + (-9.5 + 16.4i)T^{2} \)
29 \( 1 + 1.17T + 29T^{2} \)
31 \( 1 + (-1 + 1.73i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.58 - 2.74i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 10.8T + 41T^{2} \)
43 \( 1 + 11.6T + 43T^{2} \)
47 \( 1 + (4.5 + 7.79i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-5.03 + 8.72i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.828 - 1.43i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1 - 1.73i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (7.44 - 12.9i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 9.82T + 71T^{2} \)
73 \( 1 + (-5.32 + 9.22i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-5 - 8.66i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 12.8T + 83T^{2} \)
89 \( 1 + (0.585 + 1.01i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 9.65T + 97T^{2} \)
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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

−9.888696697208943093882441709616, −8.984211490665325387008222325083, −8.153604549053773623298148497353, −6.96746454060932155699721124529, −6.52663746343430022152695164907, −5.31221480315183791896259561176, −4.74615335642998972372434177939, −3.62247293266546208052506343663, −2.58932275794044946369920985847, −0.098979389376422334434527209844, 1.57090296905568442426883572116, 2.95292866031941491523617979303, 3.69038547712485867953916748692, 5.01157439280120531807596657127, 5.94195456040920592750479146965, 6.64195541541677801656383258954, 7.64533320000479416284116386928, 8.596599407320606547845783763869, 9.627063959613836449248259198910, 10.35976480964553128178003607584

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