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Properties

Label 2-966-161.10-c1-0-21
Degree $2$
Conductor $966$
Sign $0.484 + 0.874i$
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  + (0.327 − 0.945i)2-s + (0.458 + 0.888i)3-s + (−0.786 − 0.618i)4-s + (−0.614 + 0.0586i)5-s + (0.989 − 0.142i)6-s + (2.12 − 1.57i)7-s + (−0.841 + 0.540i)8-s + (−0.580 + 0.814i)9-s + (−0.145 + 0.599i)10-s + (0.580 − 0.200i)11-s + (0.189 − 0.981i)12-s + (−0.399 − 1.36i)13-s + (−0.791 − 2.52i)14-s + (−0.333 − 0.519i)15-s + (0.235 + 0.971i)16-s + (3.61 + 1.44i)17-s + ⋯
L(s)  = 1  + (0.231 − 0.668i)2-s + (0.264 + 0.513i)3-s + (−0.393 − 0.309i)4-s + (−0.274 + 0.0262i)5-s + (0.404 − 0.0580i)6-s + (0.803 − 0.594i)7-s + (−0.297 + 0.191i)8-s + (−0.193 + 0.271i)9-s + (−0.0460 + 0.189i)10-s + (0.175 − 0.0605i)11-s + (0.0546 − 0.283i)12-s + (−0.110 − 0.377i)13-s + (−0.211 − 0.674i)14-s + (−0.0861 − 0.134i)15-s + (0.0589 + 0.242i)16-s + (0.877 + 0.351i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.484 + 0.874i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.484 + 0.874i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(966\)    =    \(2 \cdot 3 \cdot 7 \cdot 23\)
Sign: $0.484 + 0.874i$
Analytic conductor: \(7.71354\)
Root analytic conductor: \(2.77732\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{966} (493, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 966,\ (\ :1/2),\ 0.484 + 0.874i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.69146 - 0.996985i\)
\(L(\frac12)\) \(\approx\) \(1.69146 - 0.996985i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.327 + 0.945i)T \)
3 \( 1 + (-0.458 - 0.888i)T \)
7 \( 1 + (-2.12 + 1.57i)T \)
23 \( 1 + (0.0117 + 4.79i)T \)
good5 \( 1 + (0.614 - 0.0586i)T + (4.90 - 0.946i)T^{2} \)
11 \( 1 + (-0.580 + 0.200i)T + (8.64 - 6.79i)T^{2} \)
13 \( 1 + (0.399 + 1.36i)T + (-10.9 + 7.02i)T^{2} \)
17 \( 1 + (-3.61 - 1.44i)T + (12.3 + 11.7i)T^{2} \)
19 \( 1 + (-3.76 + 1.50i)T + (13.7 - 13.1i)T^{2} \)
29 \( 1 + (0.716 + 4.98i)T + (-27.8 + 8.17i)T^{2} \)
31 \( 1 + (-8.57 + 0.408i)T + (30.8 - 2.94i)T^{2} \)
37 \( 1 + (-6.40 - 4.56i)T + (12.1 + 34.9i)T^{2} \)
41 \( 1 + (1.48 + 0.678i)T + (26.8 + 30.9i)T^{2} \)
43 \( 1 + (-4.17 + 6.49i)T + (-17.8 - 39.1i)T^{2} \)
47 \( 1 + (6.20 - 3.58i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-3.49 - 3.66i)T + (-2.52 + 52.9i)T^{2} \)
59 \( 1 + (0.698 + 0.169i)T + (52.4 + 27.0i)T^{2} \)
61 \( 1 + (3.40 + 1.75i)T + (35.3 + 49.6i)T^{2} \)
67 \( 1 + (0.973 + 5.05i)T + (-62.2 + 24.9i)T^{2} \)
71 \( 1 + (-6.76 - 7.80i)T + (-10.1 + 70.2i)T^{2} \)
73 \( 1 + (-6.98 + 8.87i)T + (-17.2 - 70.9i)T^{2} \)
79 \( 1 + (0.242 - 0.254i)T + (-3.75 - 78.9i)T^{2} \)
83 \( 1 + (1.67 + 3.66i)T + (-54.3 + 62.7i)T^{2} \)
89 \( 1 + (-0.104 + 2.18i)T + (-88.5 - 8.45i)T^{2} \)
97 \( 1 + (-2.07 + 4.55i)T + (-63.5 - 73.3i)T^{2} \)
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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

−10.07782157355422577158025840591, −9.258492126005389025953630529716, −8.113208735533989312005933943513, −7.74061315034808536175763831722, −6.30540121164109649565519228510, −5.19372062513269978108333016882, −4.40369490292293675074955824431, −3.59463875631000137108348435741, −2.49359813794743293688500306550, −0.981374635557093489672544840958, 1.38207249514554758825756372459, 2.83143595071766195964028485479, 3.98694543521718458117067791435, 5.11692912273125224021668650239, 5.82059299405158628190991946949, 6.87009056023487936497040058840, 7.80213125557737709653020209344, 8.105641334512201783530344896946, 9.210000656836977639792311438889, 9.835452582644797288143834550196

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