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Properties

Label 2-966-21.5-c1-0-22
Degree $2$
Conductor $966$
Sign $0.895 + 0.444i$
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  + (−0.866 − 0.5i)2-s + (−0.866 − 1.5i)3-s + (0.499 + 0.866i)4-s + (0.866 − 1.5i)5-s + 1.73i·6-s + (0.5 + 2.59i)7-s − 0.999i·8-s + (−1.5 + 2.59i)9-s + (−1.5 + 0.866i)10-s + (0.866 − 1.49i)12-s − 1.73i·13-s + (0.866 − 2.5i)14-s − 3·15-s + (−0.5 + 0.866i)16-s + (2.59 + 4.5i)17-s + (2.59 − 1.5i)18-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (−0.499 − 0.866i)3-s + (0.249 + 0.433i)4-s + (0.387 − 0.670i)5-s + 0.707i·6-s + (0.188 + 0.981i)7-s − 0.353i·8-s + (−0.5 + 0.866i)9-s + (−0.474 + 0.273i)10-s + (0.250 − 0.433i)12-s − 0.480i·13-s + (0.231 − 0.668i)14-s − 0.774·15-s + (−0.125 + 0.216i)16-s + (0.630 + 1.09i)17-s + (0.612 − 0.353i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.895 + 0.444i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.895 + 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(966\)    =    \(2 \cdot 3 \cdot 7 \cdot 23\)
Sign: $0.895 + 0.444i$
Analytic conductor: \(7.71354\)
Root analytic conductor: \(2.77732\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{966} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 966,\ (\ :1/2),\ 0.895 + 0.444i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.01799 - 0.238477i\)
\(L(\frac12)\) \(\approx\) \(1.01799 - 0.238477i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 + 0.5i)T \)
3 \( 1 + (0.866 + 1.5i)T \)
7 \( 1 + (-0.5 - 2.59i)T \)
23 \( 1 + (0.866 + 0.5i)T \)
good5 \( 1 + (-0.866 + 1.5i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 1.73iT - 13T^{2} \)
17 \( 1 + (-2.59 - 4.5i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3 - 1.73i)T + (9.5 + 16.4i)T^{2} \)
29 \( 1 - 6iT - 29T^{2} \)
31 \( 1 + (-3 + 1.73i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (2 - 3.46i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 - 4T + 43T^{2} \)
47 \( 1 + (2.59 - 4.5i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-7.79 + 4.5i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (5.19 + 9i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-12 - 6.92i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.5 + 4.33i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 15iT - 71T^{2} \)
73 \( 1 + (-10.5 + 6.06i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (4 - 6.92i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 6.92T + 83T^{2} \)
89 \( 1 + (3.46 - 6i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 10.3iT - 97T^{2} \)
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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

−9.958365145396410630788626940127, −9.021170227450687088293826037930, −8.290853957220871666043495492731, −7.70814642573710211063449076517, −6.51459236105846820909429063171, −5.67902050924829705499226361025, −5.01504153658695392158069780174, −3.27196116718546034265893205859, −2.02678563660850663516439690604, −1.12599623472628539730121699840, 0.78488720321465058855482677581, 2.66663255895280966722409705775, 3.90304057365746255171389376716, 4.92031240975501060690406989073, 5.84061067738401603450943865328, 6.80496670658953220724280009125, 7.37560578876856093853911016660, 8.522882640925337534358155456715, 9.572406811032265422297843559124, 9.977994111021820756197535328696

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