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Properties

Label 2-966-161.68-c1-0-11
Degree $2$
Conductor $966$
Sign $0.940 - 0.339i$
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 + (−0.866 + 0.5i)3-s + (−0.499 − 0.866i)4-s + (−1.52 + 2.63i)5-s − 0.999i·6-s + (−1.49 − 2.18i)7-s + 0.999·8-s + (0.499 − 0.866i)9-s + (−1.52 − 2.63i)10-s + (−2.07 + 1.19i)11-s + (0.866 + 0.499i)12-s − 4.32i·13-s + (2.63 − 0.202i)14-s − 3.04i·15-s + (−0.5 + 0.866i)16-s + (0.399 + 0.692i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.499 + 0.288i)3-s + (−0.249 − 0.433i)4-s + (−0.680 + 1.17i)5-s − 0.408i·6-s + (−0.564 − 0.825i)7-s + 0.353·8-s + (0.166 − 0.288i)9-s + (−0.481 − 0.833i)10-s + (−0.625 + 0.360i)11-s + (0.249 + 0.144i)12-s − 1.19i·13-s + (0.705 − 0.0539i)14-s − 0.785i·15-s + (−0.125 + 0.216i)16-s + (0.0969 + 0.167i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.940 - 0.339i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.940 - 0.339i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(966\)    =    \(2 \cdot 3 \cdot 7 \cdot 23\)
Sign: $0.940 - 0.339i$
Analytic conductor: \(7.71354\)
Root analytic conductor: \(2.77732\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{966} (229, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 966,\ (\ :1/2),\ 0.940 - 0.339i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.664331 + 0.116146i\)
\(L(\frac12)\) \(\approx\) \(0.664331 + 0.116146i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.866 - 0.5i)T \)
7 \( 1 + (1.49 + 2.18i)T \)
23 \( 1 + (-4.75 - 0.608i)T \)
good5 \( 1 + (1.52 - 2.63i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (2.07 - 1.19i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 4.32iT - 13T^{2} \)
17 \( 1 + (-0.399 - 0.692i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.07 + 1.86i)T + (-9.5 - 16.4i)T^{2} \)
29 \( 1 + 4.78T + 29T^{2} \)
31 \( 1 + (-3.53 + 2.04i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3.53 - 2.04i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 7.66iT - 41T^{2} \)
43 \( 1 - 3.15iT - 43T^{2} \)
47 \( 1 + (-7.12 - 4.11i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (9.49 - 5.48i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-10.6 + 6.15i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.01 + 3.49i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.06 + 2.34i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 6.20T + 71T^{2} \)
73 \( 1 + (-2.21 + 1.27i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-7.62 - 4.40i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 0.377T + 83T^{2} \)
89 \( 1 + (0.328 - 0.569i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 2.07T + 97T^{2} \)
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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

−10.11188690256050535139951717737, −9.446873773592461205071323829728, −8.062898225672445523037666016253, −7.42089364803387919624296087092, −6.85930193572448756004678763557, −5.92956787827594953999734227766, −4.92623612592023576138463413943, −3.77673239011887897707311682759, −2.87458377696956999364152899553, −0.53739097974412983753919657764, 0.879566930009062999745046639386, 2.28501890815015873230468613027, 3.60838192026336725886419829422, 4.71394911188894844384330238067, 5.46409133327381804306249165611, 6.59877463713678340440177974919, 7.65771263250012282980784173809, 8.514891139131827626363434816467, 9.100535424440369257492491837418, 9.861584220151721621048807303589

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