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Properties

Label 2-966-21.20-c1-0-37
Degree $2$
Conductor $966$
Sign $-0.649 + 0.760i$
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 + (1.18 + 1.26i)3-s − 4-s − 3.87·5-s + (1.26 − 1.18i)6-s + (0.302 + 2.62i)7-s + i·8-s + (−0.215 + 2.99i)9-s + 3.87i·10-s − 5.48i·11-s + (−1.18 − 1.26i)12-s − 4.05i·13-s + (2.62 − 0.302i)14-s + (−4.57 − 4.91i)15-s + 16-s − 2.53·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (0.681 + 0.732i)3-s − 0.5·4-s − 1.73·5-s + (0.517 − 0.481i)6-s + (0.114 + 0.993i)7-s + 0.353i·8-s + (−0.0716 + 0.997i)9-s + 1.22i·10-s − 1.65i·11-s + (−0.340 − 0.366i)12-s − 1.12i·13-s + (0.702 − 0.0809i)14-s + (−1.18 − 1.26i)15-s + 0.250·16-s − 0.615·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.649 + 0.760i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.649 + 0.760i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(966\)    =    \(2 \cdot 3 \cdot 7 \cdot 23\)
Sign: $-0.649 + 0.760i$
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: \((2,\ 966,\ (\ :1/2),\ -0.649 + 0.760i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.274803 - 0.595857i\)
\(L(\frac12)\) \(\approx\) \(0.274803 - 0.595857i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.18 - 1.26i)T \)
7 \( 1 + (-0.302 - 2.62i)T \)
23 \( 1 + iT \)
good5 \( 1 + 3.87T + 5T^{2} \)
11 \( 1 + 5.48iT - 11T^{2} \)
13 \( 1 + 4.05iT - 13T^{2} \)
17 \( 1 + 2.53T + 17T^{2} \)
19 \( 1 - 1.59iT - 19T^{2} \)
29 \( 1 + 9.78iT - 29T^{2} \)
31 \( 1 + 6.36iT - 31T^{2} \)
37 \( 1 - 3.09T + 37T^{2} \)
41 \( 1 + 2.14T + 41T^{2} \)
43 \( 1 + 6.76T + 43T^{2} \)
47 \( 1 + 0.573T + 47T^{2} \)
53 \( 1 + 6.95iT - 53T^{2} \)
59 \( 1 - 4.91T + 59T^{2} \)
61 \( 1 + 12.1iT - 61T^{2} \)
67 \( 1 + 12.4T + 67T^{2} \)
71 \( 1 - 8.75iT - 71T^{2} \)
73 \( 1 + 0.527iT - 73T^{2} \)
79 \( 1 + 1.78T + 79T^{2} \)
83 \( 1 + 0.461T + 83T^{2} \)
89 \( 1 + 12.2T + 89T^{2} \)
97 \( 1 - 8.09iT - 97T^{2} \)
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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

−9.735708167957124079879116687408, −8.702686057903138944953480583049, −8.237254896335345622646836168790, −7.82412208710177695150853500448, −6.04856410960431502201500537795, −5.03138320715357745497585228176, −4.01218184771956368158325508166, −3.34245889772136578422556091620, −2.54550839946177786135684856384, −0.29128304993551082537741797370, 1.48668471067578814825217363330, 3.25778216616696384992794708948, 4.25696871473515863834742603291, 4.69997054337006184188467958242, 6.73066199081184132983204466721, 7.14796165737700584671658898581, 7.47134767674533373025449642789, 8.501145180252194962604065249589, 9.077281453430294840782015655679, 10.20558179615522311884933428695

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