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Properties

Label 2-966-21.20-c1-0-53
Degree $2$
Conductor $966$
Sign $-0.883 + 0.467i$
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 + (−0.711 − 1.57i)3-s − 4-s + 0.334·5-s + (1.57 − 0.711i)6-s + (−0.168 − 2.64i)7-s − i·8-s + (−1.98 + 2.24i)9-s + 0.334i·10-s + 0.616i·11-s + (0.711 + 1.57i)12-s − 3.23i·13-s + (2.64 − 0.168i)14-s + (−0.237 − 0.527i)15-s + 16-s − 3.15·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (−0.410 − 0.911i)3-s − 0.5·4-s + 0.149·5-s + (0.644 − 0.290i)6-s + (−0.0637 − 0.997i)7-s − 0.353i·8-s + (−0.662 + 0.748i)9-s + 0.105i·10-s + 0.185i·11-s + (0.205 + 0.455i)12-s − 0.898i·13-s + (0.705 − 0.0450i)14-s + (−0.0613 − 0.136i)15-s + 0.250·16-s − 0.766·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.883 + 0.467i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.883 + 0.467i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(966\)    =    \(2 \cdot 3 \cdot 7 \cdot 23\)
Sign: $-0.883 + 0.467i$
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: \((2,\ 966,\ (\ :1/2),\ -0.883 + 0.467i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.103757 - 0.417700i\)
\(L(\frac12)\) \(\approx\) \(0.103757 - 0.417700i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.711 + 1.57i)T \)
7 \( 1 + (0.168 + 2.64i)T \)
23 \( 1 - iT \)
good5 \( 1 - 0.334T + 5T^{2} \)
11 \( 1 - 0.616iT - 11T^{2} \)
13 \( 1 + 3.23iT - 13T^{2} \)
17 \( 1 + 3.15T + 17T^{2} \)
19 \( 1 - 0.890iT - 19T^{2} \)
29 \( 1 + 2.14iT - 29T^{2} \)
31 \( 1 - 2.97iT - 31T^{2} \)
37 \( 1 + 2.08T + 37T^{2} \)
41 \( 1 + 5.08T + 41T^{2} \)
43 \( 1 + 11.3T + 43T^{2} \)
47 \( 1 - 2.96T + 47T^{2} \)
53 \( 1 + 0.00920iT - 53T^{2} \)
59 \( 1 + 13.1T + 59T^{2} \)
61 \( 1 + 8.92iT - 61T^{2} \)
67 \( 1 - 5.51T + 67T^{2} \)
71 \( 1 - 4.57iT - 71T^{2} \)
73 \( 1 + 6.21iT - 73T^{2} \)
79 \( 1 + 7.38T + 79T^{2} \)
83 \( 1 - 11.3T + 83T^{2} \)
89 \( 1 + 15.6T + 89T^{2} \)
97 \( 1 + 10.2iT - 97T^{2} \)
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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

−9.701078560529822369027299789092, −8.454531261984056700027417290878, −7.84391202661622327582330259919, −7.06445038258317238079437296512, −6.41532082901820211797380609623, −5.50884262813503044455401678469, −4.59233025082258813861368761530, −3.31086985040550957072321971989, −1.69277834093558783755798011693, −0.20363313381305669482445988482, 1.94861262722800559384426337396, 3.08918928373789181853519038237, 4.15211135041459082772540965073, 4.99907078643048931286286923062, 5.86779342694488667285213498804, 6.73417967728748973949470104186, 8.317407931030418799067292148118, 9.002561582722671730895532602443, 9.571038090493476683324977373574, 10.36134019986556302653932095260

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