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Properties

Label 2-966-21.20-c1-0-29
Degree $2$
Conductor $966$
Sign $-0.0403 - 0.999i$
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 + (1.66 − 0.462i)3-s − 4-s + 1.95·5-s + (0.462 + 1.66i)6-s + (−0.602 + 2.57i)7-s − i·8-s + (2.57 − 1.54i)9-s + 1.95i·10-s + 4.18i·11-s + (−1.66 + 0.462i)12-s + 7.00i·13-s + (−2.57 − 0.602i)14-s + (3.25 − 0.901i)15-s + 16-s − 3.56·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (0.963 − 0.266i)3-s − 0.5·4-s + 0.872·5-s + (0.188 + 0.681i)6-s + (−0.227 + 0.973i)7-s − 0.353i·8-s + (0.857 − 0.514i)9-s + 0.616i·10-s + 1.26i·11-s + (−0.481 + 0.133i)12-s + 1.94i·13-s + (−0.688 − 0.161i)14-s + (0.840 − 0.232i)15-s + 0.250·16-s − 0.865·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0403 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0403 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(966\)    =    \(2 \cdot 3 \cdot 7 \cdot 23\)
Sign: $-0.0403 - 0.999i$
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: \((2,\ 966,\ (\ :1/2),\ -0.0403 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.60158 + 1.66757i\)
\(L(\frac12)\) \(\approx\) \(1.60158 + 1.66757i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.66 + 0.462i)T \)
7 \( 1 + (0.602 - 2.57i)T \)
23 \( 1 + iT \)
good5 \( 1 - 1.95T + 5T^{2} \)
11 \( 1 - 4.18iT - 11T^{2} \)
13 \( 1 - 7.00iT - 13T^{2} \)
17 \( 1 + 3.56T + 17T^{2} \)
19 \( 1 + 8.08iT - 19T^{2} \)
29 \( 1 - 2.33iT - 29T^{2} \)
31 \( 1 + 0.237iT - 31T^{2} \)
37 \( 1 + 0.324T + 37T^{2} \)
41 \( 1 - 8.81T + 41T^{2} \)
43 \( 1 - 8.54T + 43T^{2} \)
47 \( 1 + 0.174T + 47T^{2} \)
53 \( 1 - 5.65iT - 53T^{2} \)
59 \( 1 + 7.74T + 59T^{2} \)
61 \( 1 + 9.18iT - 61T^{2} \)
67 \( 1 - 7.68T + 67T^{2} \)
71 \( 1 - 5.42iT - 71T^{2} \)
73 \( 1 + 11.1iT - 73T^{2} \)
79 \( 1 - 12.2T + 79T^{2} \)
83 \( 1 - 17.9T + 83T^{2} \)
89 \( 1 + 1.21T + 89T^{2} \)
97 \( 1 + 16.7iT - 97T^{2} \)
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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

−9.487839319759475711177461678823, −9.341118276030603332668241952726, −8.868833495977848379988065693655, −7.59557738642959102964691476290, −6.75534250886372237128610813748, −6.33659537812520337387550191679, −4.91357770997533557967557710605, −4.22354899799325563985911246817, −2.52594059696717004826237973457, −1.93188813644790608936114146893, 1.00846941100305524894639669769, 2.38626944463052855607680241333, 3.37309724515453161136287265733, 4.02648700464611532469742191390, 5.41677769988439846728429883260, 6.19531537835781638952903602475, 7.74420861722502812204518283571, 8.127427864798389621131508468038, 9.192799941181682712609874053412, 9.914666437419422253803946425445

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