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Properties

Label 2-966-161.45-c1-0-2
Degree $2$
Conductor $966$
Sign $-0.903 - 0.429i$
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 + (1.92 + 3.32i)5-s − 0.999i·6-s + (−0.661 − 2.56i)7-s − 0.999·8-s + (0.499 + 0.866i)9-s + (−1.92 + 3.32i)10-s + (−2.74 − 1.58i)11-s + (0.866 − 0.499i)12-s + 3.31i·13-s + (1.88 − 1.85i)14-s − 3.84i·15-s + (−0.5 − 0.866i)16-s + (−2.62 + 4.55i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.499 − 0.288i)3-s + (−0.249 + 0.433i)4-s + (0.859 + 1.48i)5-s − 0.408i·6-s + (−0.250 − 0.968i)7-s − 0.353·8-s + (0.166 + 0.288i)9-s + (−0.607 + 1.05i)10-s + (−0.826 − 0.477i)11-s + (0.249 − 0.144i)12-s + 0.919i·13-s + (0.504 − 0.495i)14-s − 0.992i·15-s + (−0.125 − 0.216i)16-s + (−0.637 + 1.10i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.903 - 0.429i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.903 - 0.429i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(966\)    =    \(2 \cdot 3 \cdot 7 \cdot 23\)
Sign: $-0.903 - 0.429i$
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: \((2,\ 966,\ (\ :1/2),\ -0.903 - 0.429i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.274731 + 1.21789i\)
\(L(\frac12)\) \(\approx\) \(0.274731 + 1.21789i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.661 + 2.56i)T \)
23 \( 1 + (3.26 + 3.51i)T \)
good5 \( 1 + (-1.92 - 3.32i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (2.74 + 1.58i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 3.31iT - 13T^{2} \)
17 \( 1 + (2.62 - 4.55i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4.13 - 7.15i)T + (-9.5 + 16.4i)T^{2} \)
29 \( 1 - 7.98T + 29T^{2} \)
31 \( 1 + (5.08 + 2.93i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (9.42 - 5.43i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 10.0iT - 41T^{2} \)
43 \( 1 - 7.75iT - 43T^{2} \)
47 \( 1 + (2.67 - 1.54i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1.30 - 0.754i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2.61 - 1.51i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.02 - 3.50i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.02 - 0.589i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 7.16T + 71T^{2} \)
73 \( 1 + (-11.7 - 6.80i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.906 - 0.523i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 1.08T + 83T^{2} \)
89 \( 1 + (7.52 + 13.0i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 5.59T + 97T^{2} \)
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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

−10.35030062736660773332358068069, −9.911701004568250818180001401849, −8.446688219250327963741816355905, −7.54014997505796600310059053147, −6.76834484631044778947255818850, −6.25930190832253771273449345908, −5.53188909012515704441613864601, −4.16152014144142643721892842538, −3.21363370779245988570979364759, −1.88669031473017234484261745495, 0.52412761465111353479382931332, 2.03603633992062355087905328618, 3.07259369209180060332176781515, 4.78098142370376024699694505786, 5.15406132910207423272978718882, 5.65821962455467421469747835727, 6.92709902677355437836619351947, 8.336112642245161842394538283054, 9.168813867872370487559168688764, 9.607377406996521546785729302536

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