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Properties

Label 2-966-161.68-c1-0-27
Degree $2$
Conductor $966$
Sign $-0.253 + 0.967i$
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 + (0.856 − 1.48i)5-s + 0.999i·6-s + (−0.871 − 2.49i)7-s + 0.999·8-s + (0.499 − 0.866i)9-s + (0.856 + 1.48i)10-s + (−4.71 + 2.72i)11-s + (−0.866 − 0.499i)12-s − 1.89i·13-s + (2.59 + 0.494i)14-s − 1.71i·15-s + (−0.5 + 0.866i)16-s + (−0.914 − 1.58i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (0.499 − 0.288i)3-s + (−0.249 − 0.433i)4-s + (0.382 − 0.663i)5-s + 0.408i·6-s + (−0.329 − 0.944i)7-s + 0.353·8-s + (0.166 − 0.288i)9-s + (0.270 + 0.468i)10-s + (−1.42 + 0.820i)11-s + (−0.249 − 0.144i)12-s − 0.525i·13-s + (0.694 + 0.132i)14-s − 0.442i·15-s + (−0.125 + 0.216i)16-s + (−0.221 − 0.384i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.253 + 0.967i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.253 + 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(966\)    =    \(2 \cdot 3 \cdot 7 \cdot 23\)
Sign: $-0.253 + 0.967i$
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: \((2,\ 966,\ (\ :1/2),\ -0.253 + 0.967i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.621716 - 0.805244i\)
\(L(\frac12)\) \(\approx\) \(0.621716 - 0.805244i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.871 + 2.49i)T \)
23 \( 1 + (-2.14 + 4.28i)T \)
good5 \( 1 + (-0.856 + 1.48i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (4.71 - 2.72i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 1.89iT - 13T^{2} \)
17 \( 1 + (0.914 + 1.58i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.07 + 1.85i)T + (-9.5 - 16.4i)T^{2} \)
29 \( 1 + 3.17T + 29T^{2} \)
31 \( 1 + (0.763 - 0.440i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (9.21 + 5.32i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 11.9iT - 41T^{2} \)
43 \( 1 + 10.3iT - 43T^{2} \)
47 \( 1 + (11.4 + 6.62i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.86 - 2.23i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-11.6 + 6.71i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.09 + 3.63i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.17 - 1.25i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 8.36T + 71T^{2} \)
73 \( 1 + (-9.32 + 5.38i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (9.82 + 5.67i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 4.05T + 83T^{2} \)
89 \( 1 + (3.76 - 6.52i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 8.00T + 97T^{2} \)
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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

−9.735376027352680747742005622114, −8.842251948282058041372610455388, −8.046178320352142984865276722471, −7.28862648646520856998287866318, −6.69306900661532536707942180441, −5.29923697646923877609100907365, −4.78565544610060156365436909551, −3.34951263076246847204511345898, −1.99600923686992754716176319746, −0.47063005490066884678797798183, 1.94637813633741290796879529071, 2.85531642911325908781518500182, 3.52549700945310223727566216336, 5.03645551364230972327109675336, 5.87905722384803849409558846672, 6.99028134537466198186878565059, 8.086185474972084578471598270933, 8.670049998906981992446342760872, 9.556142951492013441611437498352, 10.18241343766144690967888241795

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