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Properties

Label 2-966-21.20-c1-0-43
Degree $2$
Conductor $966$
Sign $0.717 - 0.696i$
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  + i·2-s + (1.55 + 0.758i)3-s − 4-s + 2.55·5-s + (−0.758 + 1.55i)6-s + (0.900 − 2.48i)7-s − i·8-s + (1.84 + 2.36i)9-s + 2.55i·10-s − 6.10i·11-s + (−1.55 − 0.758i)12-s + 4.03i·13-s + (2.48 + 0.900i)14-s + (3.97 + 1.93i)15-s + 16-s + 1.51·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (0.899 + 0.437i)3-s − 0.5·4-s + 1.14·5-s + (−0.309 + 0.635i)6-s + (0.340 − 0.940i)7-s − 0.353i·8-s + (0.616 + 0.787i)9-s + 0.807i·10-s − 1.84i·11-s + (−0.449 − 0.218i)12-s + 1.11i·13-s + (0.664 + 0.240i)14-s + (1.02 + 0.500i)15-s + 0.250·16-s + 0.367·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.717 - 0.696i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.717 - 0.696i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(966\)    =    \(2 \cdot 3 \cdot 7 \cdot 23\)
Sign: $0.717 - 0.696i$
Analytic conductor: \(7.71354\)
Root analytic conductor: \(2.77732\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{966} (461, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 966,\ (\ :1/2),\ 0.717 - 0.696i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.42549 + 0.983342i\)
\(L(\frac12)\) \(\approx\) \(2.42549 + 0.983342i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 + (-1.55 - 0.758i)T \)
7 \( 1 + (-0.900 + 2.48i)T \)
23 \( 1 - iT \)
good5 \( 1 - 2.55T + 5T^{2} \)
11 \( 1 + 6.10iT - 11T^{2} \)
13 \( 1 - 4.03iT - 13T^{2} \)
17 \( 1 - 1.51T + 17T^{2} \)
19 \( 1 + 4.48iT - 19T^{2} \)
29 \( 1 - 6.55iT - 29T^{2} \)
31 \( 1 + 4.67iT - 31T^{2} \)
37 \( 1 - 9.05T + 37T^{2} \)
41 \( 1 + 6.16T + 41T^{2} \)
43 \( 1 + 3.13T + 43T^{2} \)
47 \( 1 + 0.328T + 47T^{2} \)
53 \( 1 - 13.7iT - 53T^{2} \)
59 \( 1 + 13.6T + 59T^{2} \)
61 \( 1 + 3.68iT - 61T^{2} \)
67 \( 1 - 6.92T + 67T^{2} \)
71 \( 1 - 11.8iT - 71T^{2} \)
73 \( 1 - 6.01iT - 73T^{2} \)
79 \( 1 + 4.13T + 79T^{2} \)
83 \( 1 + 13.6T + 83T^{2} \)
89 \( 1 - 5.12T + 89T^{2} \)
97 \( 1 + 4.80iT - 97T^{2} \)
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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

−9.838391364466260012352740442101, −9.185336927637595082649210043797, −8.554684807492230663641253574986, −7.66600351439565397328976208662, −6.75612390491953675549270330725, −5.86644677153824060928423692478, −4.87775598838837170465807911477, −3.92755832751493217253425022406, −2.85360022012273819449952986938, −1.35021411125497121946756988241, 1.63665959567481698336379695341, 2.19108174400968047922732946797, 3.16145801242642729245319151441, 4.53615091077502692285419736159, 5.51933974331264344682227615976, 6.44242977712039880366145075304, 7.71409588297941623379135523921, 8.283017423161857812588376166085, 9.371620697130385523690753257982, 9.834451857178498883196958608196

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