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Properties

Label 2-966-161.68-c1-0-23
Degree $2$
Conductor $966$
Sign $-0.713 + 0.700i$
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 + (−1.70 + 2.95i)5-s − 0.999i·6-s + (−2.64 − 0.121i)7-s − 0.999·8-s + (0.499 − 0.866i)9-s + (1.70 + 2.95i)10-s + (2.85 − 1.64i)11-s + (−0.866 − 0.499i)12-s − 6.36i·13-s + (−1.42 + 2.22i)14-s + 3.40i·15-s + (−0.5 + 0.866i)16-s + (−0.645 − 1.11i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (0.499 − 0.288i)3-s + (−0.249 − 0.433i)4-s + (−0.762 + 1.31i)5-s − 0.408i·6-s + (−0.998 − 0.0457i)7-s − 0.353·8-s + (0.166 − 0.288i)9-s + (0.538 + 0.933i)10-s + (0.860 − 0.496i)11-s + (−0.249 − 0.144i)12-s − 1.76i·13-s + (−0.381 + 0.595i)14-s + 0.879i·15-s + (−0.125 + 0.216i)16-s + (−0.156 − 0.271i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.713 + 0.700i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.713 + 0.700i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(966\)    =    \(2 \cdot 3 \cdot 7 \cdot 23\)
Sign: $-0.713 + 0.700i$
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: \((2,\ 966,\ (\ :1/2),\ -0.713 + 0.700i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.490975 - 1.20183i\)
\(L(\frac12)\) \(\approx\) \(0.490975 - 1.20183i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.64 + 0.121i)T \)
23 \( 1 + (3.89 + 2.79i)T \)
good5 \( 1 + (1.70 - 2.95i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-2.85 + 1.64i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 6.36iT - 13T^{2} \)
17 \( 1 + (0.645 + 1.11i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.86 + 4.96i)T + (-9.5 - 16.4i)T^{2} \)
29 \( 1 - 6.29T + 29T^{2} \)
31 \( 1 + (7.37 - 4.26i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.89 + 2.82i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 1.07iT - 41T^{2} \)
43 \( 1 + 5.21iT - 43T^{2} \)
47 \( 1 + (7.71 + 4.45i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (4.88 - 2.82i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4.68 + 2.70i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.51 - 2.62i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-13.9 + 8.04i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 12.7T + 71T^{2} \)
73 \( 1 + (13.5 - 7.80i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-4.73 - 2.73i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 13.3T + 83T^{2} \)
89 \( 1 + (5.20 - 9.00i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 6.95T + 97T^{2} \)
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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

−9.918146822946080967352132525624, −8.951268578567228532831104907142, −8.019985558063285891026915848252, −6.98546046500694673964108048207, −6.52270662036077408508721343706, −5.32018449015585825745431523715, −3.72813847001200699262760592299, −3.30224203815157473398011471112, −2.55751899094515390317437419137, −0.50746153178967865130674145620, 1.67928275450393193714903503011, 3.58853938226193441553711489083, 4.08268884176564764666928889075, 4.89160352603680631102734377100, 6.11477462763566268300497641897, 6.94107457819566326360855957809, 7.900796674643812851855961087516, 8.653499224806793128858908485094, 9.428832992872322323301617847819, 9.775858851226398502881594911118

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