Properties

Label 2-966-7.2-c1-0-4
Degree $2$
Conductor $966$
Sign $0.469 - 0.883i$
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 + (−0.5 + 0.866i)5-s + 0.999·6-s + (−2.25 − 1.38i)7-s − 0.999·8-s + (−0.499 + 0.866i)9-s + (0.499 + 0.866i)10-s + (1.82 + 3.16i)11-s + (0.499 − 0.866i)12-s − 0.851·13-s + (−2.32 + 1.25i)14-s − 0.999·15-s + (−0.5 + 0.866i)16-s + (3.08 + 5.33i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (0.288 + 0.499i)3-s + (−0.249 − 0.433i)4-s + (−0.223 + 0.387i)5-s + 0.408·6-s + (−0.851 − 0.524i)7-s − 0.353·8-s + (−0.166 + 0.288i)9-s + (0.158 + 0.273i)10-s + (0.550 + 0.954i)11-s + (0.144 − 0.249i)12-s − 0.236·13-s + (−0.622 + 0.336i)14-s − 0.258·15-s + (−0.125 + 0.216i)16-s + (0.747 + 1.29i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.469 - 0.883i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.469 - 0.883i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(966\)    =    \(2 \cdot 3 \cdot 7 \cdot 23\)
Sign: $0.469 - 0.883i$
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: \((2,\ 966,\ (\ :1/2),\ 0.469 - 0.883i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.25396 + 0.753839i\)
\(L(\frac12)\) \(\approx\) \(1.25396 + 0.753839i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.25 + 1.38i)T \)
23 \( 1 + (-0.5 + 0.866i)T \)
good5 \( 1 + (0.5 - 0.866i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.82 - 3.16i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 0.851T + 13T^{2} \)
17 \( 1 + (-3.08 - 5.33i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.75 - 4.76i)T + (-9.5 - 16.4i)T^{2} \)
29 \( 1 - 4.14T + 29T^{2} \)
31 \( 1 + (1.67 + 2.90i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (3.40 - 5.90i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 2.49T + 41T^{2} \)
43 \( 1 - 7.30T + 43T^{2} \)
47 \( 1 + (1.17 - 2.04i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (3.15 + 5.46i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.58 + 2.73i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5 - 8.66i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.17 - 8.97i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 8.45T + 71T^{2} \)
73 \( 1 + (0.672 + 1.16i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-5.72 + 9.92i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 7.65T + 83T^{2} \)
89 \( 1 + (-3.40 + 5.90i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 0.444T + 97T^{2} \)
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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

−10.22044550402119103778244458348, −9.671211242014925382463957996382, −8.685603098069452220235783201888, −7.64810800099080981168514704514, −6.67993786917609321301730471184, −5.81920480692432619269436352399, −4.50019237711015107665076555007, −3.81665551182620516028775195362, −3.03593217699854680678221819539, −1.63133496660043055626531320438, 0.59308165956348605226966844169, 2.63909438833366448904268279161, 3.44216289169571614073568948012, 4.70970745490661300677178128648, 5.66799099443815850577179651877, 6.54035565591304201360721842506, 7.17611299036970457531540246761, 8.213601268638858256368773469820, 8.974395614987712120416913079545, 9.425269981941165787275217563848

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