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Properties

Label 2-966-7.4-c1-0-6
Degree $2$
Conductor $966$
Sign $-0.832 - 0.553i$
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 + (1.44 + 2.50i)5-s + 0.999·6-s + (−1.32 + 2.29i)7-s − 0.999·8-s + (−0.499 − 0.866i)9-s + (−1.44 + 2.50i)10-s + (−0.809 + 1.40i)11-s + (0.499 + 0.866i)12-s − 3.02·13-s − 2.64·14-s + 2.89·15-s + (−0.5 − 0.866i)16-s + (−2.77 + 4.80i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (0.288 − 0.499i)3-s + (−0.249 + 0.433i)4-s + (0.647 + 1.12i)5-s + 0.408·6-s + (−0.499 + 0.866i)7-s − 0.353·8-s + (−0.166 − 0.288i)9-s + (−0.458 + 0.793i)10-s + (−0.244 + 0.422i)11-s + (0.144 + 0.249i)12-s − 0.839·13-s − 0.707·14-s + 0.748·15-s + (−0.125 − 0.216i)16-s + (−0.672 + 1.16i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.832 - 0.553i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.832 - 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(966\)    =    \(2 \cdot 3 \cdot 7 \cdot 23\)
Sign: $-0.832 - 0.553i$
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: \((2,\ 966,\ (\ :1/2),\ -0.832 - 0.553i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.469174 + 1.55245i\)
\(L(\frac12)\) \(\approx\) \(0.469174 + 1.55245i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.32 - 2.29i)T \)
23 \( 1 + (0.5 + 0.866i)T \)
good5 \( 1 + (-1.44 - 2.50i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (0.809 - 1.40i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 3.02T + 13T^{2} \)
17 \( 1 + (2.77 - 4.80i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.38 + 2.39i)T + (-9.5 + 16.4i)T^{2} \)
29 \( 1 - 1.76T + 29T^{2} \)
31 \( 1 + (-0.802 + 1.39i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.513 - 0.888i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 6.76T + 41T^{2} \)
43 \( 1 - 2T + 43T^{2} \)
47 \( 1 + (-2.07 - 3.59i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.44 - 4.24i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.47 + 6.02i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.89 - 11.9i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.61 - 6.26i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 4.87T + 71T^{2} \)
73 \( 1 + (2.56 - 4.44i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3.44 - 5.96i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 5.15T + 83T^{2} \)
89 \( 1 + (-7.28 - 12.6i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 14.6T + 97T^{2} \)
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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

−10.23178931633660148277714660057, −9.441174364397450936415982970079, −8.606314205882077723286693416118, −7.65362376891665782171404997178, −6.74775498431318438693629793867, −6.31381028336735837583852585175, −5.44151960059094427678715550835, −4.15678081309367158709916973171, −2.75300824349481686765708202252, −2.29436673472892591795686354745, 0.60860618292638880827794247229, 2.14457874918099353107650352071, 3.29153890395277337995721667323, 4.41387616499467213465800124445, 4.99633704297650228558182586286, 5.95414155547304254494066833063, 7.13768576301867789748848257847, 8.247508806738398815116832534631, 9.224531445299627639048722894567, 9.641355957438975510669883245112

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