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Properties

Label 2-966-161.45-c1-0-29
Degree $2$
Conductor $966$
Sign $-0.287 - 0.957i$
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 + (−1.81 − 3.14i)5-s + 0.999i·6-s + (−1.58 − 2.11i)7-s + 0.999·8-s + (0.499 + 0.866i)9-s + (−1.81 + 3.14i)10-s + (0.401 + 0.231i)11-s + (0.866 − 0.499i)12-s − 6.88i·13-s + (−1.04 + 2.43i)14-s + 3.63i·15-s + (−0.5 − 0.866i)16-s + (1.54 − 2.68i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.499 − 0.288i)3-s + (−0.249 + 0.433i)4-s + (−0.812 − 1.40i)5-s + 0.408i·6-s + (−0.598 − 0.800i)7-s + 0.353·8-s + (0.166 + 0.288i)9-s + (−0.574 + 0.995i)10-s + (0.121 + 0.0699i)11-s + (0.249 − 0.144i)12-s − 1.90i·13-s + (−0.278 + 0.649i)14-s + 0.938i·15-s + (−0.125 − 0.216i)16-s + (0.375 − 0.649i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.287 - 0.957i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.287 - 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(966\)    =    \(2 \cdot 3 \cdot 7 \cdot 23\)
Sign: $-0.287 - 0.957i$
Analytic conductor: \(7.71354\)
Root analytic conductor: \(2.77732\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{966} (367, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 966,\ (\ :1/2),\ -0.287 - 0.957i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.295253 + 0.396879i\)
\(L(\frac12)\) \(\approx\) \(0.295253 + 0.396879i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.58 + 2.11i)T \)
23 \( 1 + (2.20 - 4.25i)T \)
good5 \( 1 + (1.81 + 3.14i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.401 - 0.231i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 6.88iT - 13T^{2} \)
17 \( 1 + (-1.54 + 2.68i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.90 + 5.02i)T + (-9.5 + 16.4i)T^{2} \)
29 \( 1 - 3.72T + 29T^{2} \)
31 \( 1 + (2.45 + 1.41i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-6.15 + 3.55i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 8.56iT - 41T^{2} \)
43 \( 1 + 2.69iT - 43T^{2} \)
47 \( 1 + (2.16 - 1.24i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-4.36 - 2.51i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-9.36 - 5.40i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.47 - 2.55i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.25 + 2.45i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 8.05T + 71T^{2} \)
73 \( 1 + (7.09 + 4.09i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (9.37 - 5.41i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 0.215T + 83T^{2} \)
89 \( 1 + (5.36 + 9.29i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 12.0T + 97T^{2} \)
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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

−9.547753262312270662040687858983, −8.581788106649190454108207793092, −7.82104189093581464093053656794, −7.21963491629532640741615254169, −5.81540594906156048340260675196, −4.86888432271740035019962183082, −4.04683327068329929331788550097, −2.92972358727500172566516224760, −1.02399818425044024716041223504, −0.34657035581856785770032735838, 2.16806060606369803741975755813, 3.60599498892022211191440330358, 4.34982965389906593499238209793, 5.82438075741588575599232237955, 6.52085703587985626266259713984, 6.92304359554682527610012210827, 8.101707096813707300719576653644, 8.857241287444508295717099970559, 9.903815066829298264316685106534, 10.42754427913425112864405660887

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