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Properties

Label 2-966-21.20-c1-0-36
Degree $2$
Conductor $966$
Sign $0.943 - 0.330i$
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 + (−0.950 + 1.44i)3-s − 4-s + 0.575·5-s + (−1.44 − 0.950i)6-s + (2.10 + 1.60i)7-s − i·8-s + (−1.19 − 2.75i)9-s + 0.575i·10-s − 6.32i·11-s + (0.950 − 1.44i)12-s − 2.57i·13-s + (−1.60 + 2.10i)14-s + (−0.547 + 0.834i)15-s + 16-s + 6.33·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (−0.548 + 0.836i)3-s − 0.5·4-s + 0.257·5-s + (−0.591 − 0.387i)6-s + (0.793 + 0.608i)7-s − 0.353i·8-s + (−0.398 − 0.917i)9-s + 0.182i·10-s − 1.90i·11-s + (0.274 − 0.418i)12-s − 0.714i·13-s + (−0.429 + 0.561i)14-s + (−0.141 + 0.215i)15-s + 0.250·16-s + 1.53·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.943 - 0.330i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.943 - 0.330i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(966\)    =    \(2 \cdot 3 \cdot 7 \cdot 23\)
Sign: $0.943 - 0.330i$
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: \((2,\ 966,\ (\ :1/2),\ 0.943 - 0.330i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.26270 + 0.214484i\)
\(L(\frac12)\) \(\approx\) \(1.26270 + 0.214484i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.950 - 1.44i)T \)
7 \( 1 + (-2.10 - 1.60i)T \)
23 \( 1 + iT \)
good5 \( 1 - 0.575T + 5T^{2} \)
11 \( 1 + 6.32iT - 11T^{2} \)
13 \( 1 + 2.57iT - 13T^{2} \)
17 \( 1 - 6.33T + 17T^{2} \)
19 \( 1 + 4.31iT - 19T^{2} \)
29 \( 1 + 10.1iT - 29T^{2} \)
31 \( 1 + 1.82iT - 31T^{2} \)
37 \( 1 + 3.23T + 37T^{2} \)
41 \( 1 + 10.6T + 41T^{2} \)
43 \( 1 - 6.83T + 43T^{2} \)
47 \( 1 - 1.58T + 47T^{2} \)
53 \( 1 - 4.99iT - 53T^{2} \)
59 \( 1 + 2.62T + 59T^{2} \)
61 \( 1 - 6.43iT - 61T^{2} \)
67 \( 1 + 1.55T + 67T^{2} \)
71 \( 1 + 4.59iT - 71T^{2} \)
73 \( 1 - 13.0iT - 73T^{2} \)
79 \( 1 - 16.7T + 79T^{2} \)
83 \( 1 - 2.97T + 83T^{2} \)
89 \( 1 - 3.91T + 89T^{2} \)
97 \( 1 + 17.9iT - 97T^{2} \)
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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

−10.01150054797421824362719764117, −9.148516962939841462839571472573, −8.384540591547093233386575550009, −7.74579961405638034414227269717, −6.22963748987407420006070251447, −5.69368723143553897526341871259, −5.17970209534422547897833391828, −3.95549573411333806922356335505, −2.91547399551081518374988435689, −0.69853361620532613458978728612, 1.48354020946713119060383841713, 1.90400759339505292794490171290, 3.62354359477210596109103871198, 4.80360171439093749345022613751, 5.40339704754178308133316788902, 6.73715830969200436368176843552, 7.47801967629674234296139399572, 8.108913371434217547033370102000, 9.377250291113432554245956149081, 10.21987633219916308076244742380

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