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Properties

Label 2-966-21.20-c1-0-23
Degree $2$
Conductor $966$
Sign $-0.654 - 0.756i$
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 + (1.37 + 1.05i)3-s − 4-s + 2.38·5-s + (−1.05 + 1.37i)6-s + (−2.59 − 0.527i)7-s − i·8-s + (0.764 + 2.90i)9-s + 2.38i·10-s + 4.24i·11-s + (−1.37 − 1.05i)12-s − 0.530i·13-s + (0.527 − 2.59i)14-s + (3.27 + 2.52i)15-s + 16-s + 2.11·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (0.792 + 0.610i)3-s − 0.5·4-s + 1.06·5-s + (−0.431 + 0.560i)6-s + (−0.979 − 0.199i)7-s − 0.353i·8-s + (0.254 + 0.966i)9-s + 0.754i·10-s + 1.28i·11-s + (−0.396 − 0.305i)12-s − 0.147i·13-s + (0.141 − 0.692i)14-s + (0.845 + 0.651i)15-s + 0.250·16-s + 0.512·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.654 - 0.756i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.654 - 0.756i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(966\)    =    \(2 \cdot 3 \cdot 7 \cdot 23\)
Sign: $-0.654 - 0.756i$
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: \((2,\ 966,\ (\ :1/2),\ -0.654 - 0.756i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.848765 + 1.85709i\)
\(L(\frac12)\) \(\approx\) \(0.848765 + 1.85709i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.37 - 1.05i)T \)
7 \( 1 + (2.59 + 0.527i)T \)
23 \( 1 - iT \)
good5 \( 1 - 2.38T + 5T^{2} \)
11 \( 1 - 4.24iT - 11T^{2} \)
13 \( 1 + 0.530iT - 13T^{2} \)
17 \( 1 - 2.11T + 17T^{2} \)
19 \( 1 - 2.73iT - 19T^{2} \)
29 \( 1 + 4.13iT - 29T^{2} \)
31 \( 1 - 8.63iT - 31T^{2} \)
37 \( 1 - 1.99T + 37T^{2} \)
41 \( 1 + 8.92T + 41T^{2} \)
43 \( 1 - 8.94T + 43T^{2} \)
47 \( 1 - 5.75T + 47T^{2} \)
53 \( 1 + 2.38iT - 53T^{2} \)
59 \( 1 - 4.97T + 59T^{2} \)
61 \( 1 + 3.14iT - 61T^{2} \)
67 \( 1 + 7.15T + 67T^{2} \)
71 \( 1 + 11.1iT - 71T^{2} \)
73 \( 1 + 5.68iT - 73T^{2} \)
79 \( 1 - 10.5T + 79T^{2} \)
83 \( 1 - 1.92T + 83T^{2} \)
89 \( 1 + 17.8T + 89T^{2} \)
97 \( 1 + 9.07iT - 97T^{2} \)
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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

−9.973671311406417151399002786982, −9.582210727346618859183074765424, −8.810604581754730260493902377015, −7.74960734052686505862417537251, −7.00353589546907356725426927326, −6.02978674535135223399613656368, −5.16963853572757764483326440379, −4.15154724248139104531826428681, −3.11163155545499142955722880309, −1.87297737913224253989657932169, 0.881356164598859780316037321475, 2.28126309140040462468772824931, 2.99237220200440368426047469335, 3.93246939211672484519349826230, 5.59803835218666317245139550670, 6.17734871017940263927415360934, 7.17808291496694111859979738692, 8.345077319190332239970576953285, 9.072687495283290780443629952697, 9.597077404291128135440240896270

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