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Properties

Label 2-966-161.68-c1-0-13
Degree $2$
Conductor $966$
Sign $0.998 - 0.0578i$
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 + (−0.139 + 0.240i)5-s + 0.999i·6-s + (2.42 + 1.05i)7-s − 0.999·8-s + (0.499 − 0.866i)9-s + (0.139 + 0.240i)10-s + (1.25 − 0.726i)11-s + (0.866 + 0.499i)12-s + 5.02i·13-s + (2.12 − 1.57i)14-s − 0.278i·15-s + (−0.5 + 0.866i)16-s + (−2.48 − 4.31i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.499 + 0.288i)3-s + (−0.249 − 0.433i)4-s + (−0.0622 + 0.107i)5-s + 0.408i·6-s + (0.917 + 0.397i)7-s − 0.353·8-s + (0.166 − 0.288i)9-s + (0.0439 + 0.0762i)10-s + (0.379 − 0.219i)11-s + (0.249 + 0.144i)12-s + 1.39i·13-s + (0.567 − 0.421i)14-s − 0.0718i·15-s + (−0.125 + 0.216i)16-s + (−0.603 − 1.04i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0578i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 - 0.0578i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(966\)    =    \(2 \cdot 3 \cdot 7 \cdot 23\)
Sign: $0.998 - 0.0578i$
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: \((2,\ 966,\ (\ :1/2),\ 0.998 - 0.0578i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.69257 + 0.0490055i\)
\(L(\frac12)\) \(\approx\) \(1.69257 + 0.0490055i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.42 - 1.05i)T \)
23 \( 1 + (-2.68 - 3.97i)T \)
good5 \( 1 + (0.139 - 0.240i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.25 + 0.726i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 5.02iT - 13T^{2} \)
17 \( 1 + (2.48 + 4.31i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.02 - 1.78i)T + (-9.5 - 16.4i)T^{2} \)
29 \( 1 - 5.24T + 29T^{2} \)
31 \( 1 + (-0.516 + 0.298i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.31 - 1.33i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 3.01iT - 41T^{2} \)
43 \( 1 + 6.84iT - 43T^{2} \)
47 \( 1 + (-8.05 - 4.65i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-8.65 + 4.99i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2.40 + 1.38i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.995 + 1.72i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.81 + 3.35i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 0.995T + 71T^{2} \)
73 \( 1 + (13.8 - 7.97i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.24 - 1.29i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 8.53T + 83T^{2} \)
89 \( 1 + (-3.83 + 6.64i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 4.93T + 97T^{2} \)
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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

−10.17679177909774315824445244935, −9.189534912019513040946341860206, −8.743098195943475414755429330244, −7.35074744364931889501733266394, −6.51106222190811082492279633253, −5.42492310186356497838375133678, −4.70362330199796541534190130760, −3.88400232257893272678994108761, −2.52333876135118454691986337575, −1.29851205681642380140396029923, 0.891693080468038501806957498392, 2.56162129440949025587173329325, 4.11941779063941802543394114199, 4.77692123462913188663588248035, 5.72709652046340847920179563299, 6.58192294687455437516373606440, 7.38691224867446932205834810885, 8.274883326396336655112101910842, 8.772529475573559767542757608168, 10.34282157966430039886695110898

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