Properties

Label 2-966-21.20-c1-0-12
Degree $2$
Conductor $966$
Sign $0.990 - 0.134i$
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  i·2-s + (0.952 − 1.44i)3-s − 4-s − 3.39·5-s + (−1.44 − 0.952i)6-s + (1.73 + 1.99i)7-s + i·8-s + (−1.18 − 2.75i)9-s + 3.39i·10-s + 1.55i·11-s + (−0.952 + 1.44i)12-s + 6.18i·13-s + (1.99 − 1.73i)14-s + (−3.23 + 4.91i)15-s + 16-s + 2.89·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (0.549 − 0.835i)3-s − 0.5·4-s − 1.51·5-s + (−0.590 − 0.388i)6-s + (0.656 + 0.753i)7-s + 0.353i·8-s + (−0.395 − 0.918i)9-s + 1.07i·10-s + 0.469i·11-s + (−0.274 + 0.417i)12-s + 1.71i·13-s + (0.533 − 0.464i)14-s + (−0.834 + 1.26i)15-s + 0.250·16-s + 0.701·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 - 0.134i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.990 - 0.134i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(966\)    =    \(2 \cdot 3 \cdot 7 \cdot 23\)
Sign: $0.990 - 0.134i$
Analytic conductor: \(7.71354\)
Root analytic conductor: \(2.77732\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{966} (461, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 966,\ (\ :1/2),\ 0.990 - 0.134i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.13733 + 0.0767517i\)
\(L(\frac12)\) \(\approx\) \(1.13733 + 0.0767517i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
3 \( 1 + (-0.952 + 1.44i)T \)
7 \( 1 + (-1.73 - 1.99i)T \)
23 \( 1 + iT \)
good5 \( 1 + 3.39T + 5T^{2} \)
11 \( 1 - 1.55iT - 11T^{2} \)
13 \( 1 - 6.18iT - 13T^{2} \)
17 \( 1 - 2.89T + 17T^{2} \)
19 \( 1 - 6.93iT - 19T^{2} \)
29 \( 1 - 1.06iT - 29T^{2} \)
31 \( 1 + 7.18iT - 31T^{2} \)
37 \( 1 + 6.46T + 37T^{2} \)
41 \( 1 - 4.43T + 41T^{2} \)
43 \( 1 - 7.07T + 43T^{2} \)
47 \( 1 + 11.3T + 47T^{2} \)
53 \( 1 - 2.78iT - 53T^{2} \)
59 \( 1 - 9.07T + 59T^{2} \)
61 \( 1 - 13.7iT - 61T^{2} \)
67 \( 1 - 10.9T + 67T^{2} \)
71 \( 1 - 5.72iT - 71T^{2} \)
73 \( 1 - 11.2iT - 73T^{2} \)
79 \( 1 - 0.738T + 79T^{2} \)
83 \( 1 + 2.20T + 83T^{2} \)
89 \( 1 + 3.01T + 89T^{2} \)
97 \( 1 + 6.59iT - 97T^{2} \)
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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

−9.953176623152558604045381757412, −9.046795269784265497326906687604, −8.310092953043501888558024446371, −7.76488570105304595651355151227, −6.92437458587252039488946556726, −5.67780397907016795860618029977, −4.30058953467650171988299927624, −3.73301126629794527266660169619, −2.42480261519349553613123563684, −1.42403288452063724345913999798, 0.54086654972319982021846223015, 3.13557983775912240748154156404, 3.71326619548826435449274393007, 4.78630368570166321864732137773, 5.30098816815399696205183245067, 6.91609749771095653956985831067, 7.81906457583475222182268772004, 8.084446984199272286613464785891, 8.864549735254754533868514067460, 9.996205053056493592926572687776

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