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Properties

Label 2-966-161.151-c1-0-4
Degree $2$
Conductor $966$
Sign $0.194 - 0.980i$
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 + (−0.888 + 0.458i)3-s + (−0.786 − 0.618i)4-s + (−4.06 + 0.387i)5-s + (−0.142 − 0.989i)6-s + (−0.441 − 2.60i)7-s + (0.841 − 0.540i)8-s + (0.580 − 0.814i)9-s + (0.961 − 3.96i)10-s + (−0.312 − 0.902i)11-s + (0.981 + 0.189i)12-s + (−1.52 + 0.448i)13-s + (2.60 + 0.436i)14-s + (3.43 − 2.20i)15-s + (0.235 + 0.971i)16-s + (−2.07 − 0.832i)17-s + ⋯
L(s)  = 1  + (−0.231 + 0.668i)2-s + (−0.513 + 0.264i)3-s + (−0.393 − 0.309i)4-s + (−1.81 + 0.173i)5-s + (−0.0580 − 0.404i)6-s + (−0.166 − 0.985i)7-s + (0.297 − 0.191i)8-s + (0.193 − 0.271i)9-s + (0.304 − 1.25i)10-s + (−0.0941 − 0.272i)11-s + (0.283 + 0.0546i)12-s + (−0.423 + 0.124i)13-s + (0.697 + 0.116i)14-s + (0.886 − 0.569i)15-s + (0.0589 + 0.242i)16-s + (−0.504 − 0.201i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.194 - 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.194 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(966\)    =    \(2 \cdot 3 \cdot 7 \cdot 23\)
Sign: $0.194 - 0.980i$
Analytic conductor: \(7.71354\)
Root analytic conductor: \(2.77732\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{966} (151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 966,\ (\ :1/2),\ 0.194 - 0.980i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.378518 + 0.310877i\)
\(L(\frac12)\) \(\approx\) \(0.378518 + 0.310877i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.888 - 0.458i)T \)
7 \( 1 + (0.441 + 2.60i)T \)
23 \( 1 + (-4.68 - 1.03i)T \)
good5 \( 1 + (4.06 - 0.387i)T + (4.90 - 0.946i)T^{2} \)
11 \( 1 + (0.312 + 0.902i)T + (-8.64 + 6.79i)T^{2} \)
13 \( 1 + (1.52 - 0.448i)T + (10.9 - 7.02i)T^{2} \)
17 \( 1 + (2.07 + 0.832i)T + (12.3 + 11.7i)T^{2} \)
19 \( 1 + (3.26 - 1.30i)T + (13.7 - 13.1i)T^{2} \)
29 \( 1 + (-1.00 - 6.99i)T + (-27.8 + 8.17i)T^{2} \)
31 \( 1 + (0.413 + 8.67i)T + (-30.8 + 2.94i)T^{2} \)
37 \( 1 + (-2.17 + 3.05i)T + (-12.1 - 34.9i)T^{2} \)
41 \( 1 + (2.95 - 6.48i)T + (-26.8 - 30.9i)T^{2} \)
43 \( 1 + (-2.27 - 1.46i)T + (17.8 + 39.1i)T^{2} \)
47 \( 1 + (-5.17 - 8.96i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (5.29 - 5.04i)T + (2.52 - 52.9i)T^{2} \)
59 \( 1 + (-0.359 + 1.48i)T + (-52.4 - 27.0i)T^{2} \)
61 \( 1 + (-12.2 - 6.30i)T + (35.3 + 49.6i)T^{2} \)
67 \( 1 + (4.64 - 0.894i)T + (62.2 - 24.9i)T^{2} \)
71 \( 1 + (-1.32 - 1.52i)T + (-10.1 + 70.2i)T^{2} \)
73 \( 1 + (-0.310 - 0.243i)T + (17.2 + 70.9i)T^{2} \)
79 \( 1 + (1.45 + 1.39i)T + (3.75 + 78.9i)T^{2} \)
83 \( 1 + (4.32 + 9.46i)T + (-54.3 + 62.7i)T^{2} \)
89 \( 1 + (-0.370 + 7.77i)T + (-88.5 - 8.45i)T^{2} \)
97 \( 1 + (-1.10 + 2.41i)T + (-63.5 - 73.3i)T^{2} \)
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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

−10.32906864623104335912519830966, −9.246184179321676279685922560125, −8.349695587914940927131672059275, −7.46482069610693339870557737477, −7.09958382378820417376494720173, −6.10093162773917941029214805665, −4.68824241854068390348374625169, −4.25579266157053106009620770795, −3.22961343253985154903710282620, −0.70947991364964068211393086837, 0.45306894453917296070755862098, 2.27405509478039013377318923206, 3.42153291196569268804352218091, 4.47677403975053528045101146193, 5.18613745454103771743041199224, 6.63416846028794077751429686274, 7.37895289246248067745179575989, 8.436523285786321977738587110406, 8.734068656961645311677360905797, 9.982001926890560449462081331720

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