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Properties

Label 2-966-69.68-c1-0-19
Degree $2$
Conductor $966$
Sign $0.818 - 0.574i$
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 + (1.59 − 0.673i)3-s − 4-s − 2.44·5-s + (0.673 + 1.59i)6-s + i·7-s − i·8-s + (2.09 − 2.14i)9-s − 2.44i·10-s + 1.24·11-s + (−1.59 + 0.673i)12-s + 4.09·13-s − 14-s + (−3.90 + 1.64i)15-s + 16-s + 0.678·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (0.921 − 0.388i)3-s − 0.5·4-s − 1.09·5-s + (0.274 + 0.651i)6-s + 0.377i·7-s − 0.353i·8-s + (0.697 − 0.716i)9-s − 0.773i·10-s + 0.375·11-s + (−0.460 + 0.194i)12-s + 1.13·13-s − 0.267·14-s + (−1.00 + 0.425i)15-s + 0.250·16-s + 0.164·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.818 - 0.574i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.818 - 0.574i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(966\)    =    \(2 \cdot 3 \cdot 7 \cdot 23\)
Sign: $0.818 - 0.574i$
Analytic conductor: \(7.71354\)
Root analytic conductor: \(2.77732\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{966} (827, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 966,\ (\ :1/2),\ 0.818 - 0.574i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.79043 + 0.565305i\)
\(L(\frac12)\) \(\approx\) \(1.79043 + 0.565305i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.59 + 0.673i)T \)
7 \( 1 - iT \)
23 \( 1 + (-4.68 + 1.01i)T \)
good5 \( 1 + 2.44T + 5T^{2} \)
11 \( 1 - 1.24T + 11T^{2} \)
13 \( 1 - 4.09T + 13T^{2} \)
17 \( 1 - 0.678T + 17T^{2} \)
19 \( 1 - 1.06iT - 19T^{2} \)
29 \( 1 - 0.121iT - 29T^{2} \)
31 \( 1 - 9.43T + 31T^{2} \)
37 \( 1 + 4.01iT - 37T^{2} \)
41 \( 1 - 6.95iT - 41T^{2} \)
43 \( 1 - 10.8iT - 43T^{2} \)
47 \( 1 - 6.85iT - 47T^{2} \)
53 \( 1 - 10.1T + 53T^{2} \)
59 \( 1 + 4.05iT - 59T^{2} \)
61 \( 1 + 7.68iT - 61T^{2} \)
67 \( 1 + 7.06iT - 67T^{2} \)
71 \( 1 + 5.06iT - 71T^{2} \)
73 \( 1 - 2.93T + 73T^{2} \)
79 \( 1 - 2.85iT - 79T^{2} \)
83 \( 1 + 3.55T + 83T^{2} \)
89 \( 1 - 1.72T + 89T^{2} \)
97 \( 1 + 3.78iT - 97T^{2} \)
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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

−9.734809650595217651030425990305, −8.982937420950624286310164015859, −8.187460605029499037305217299845, −7.85325509067403999204184496156, −6.77641390779879224255563680158, −6.13007361223823877468865451007, −4.66581395268288778765295535905, −3.81984480244356205865199762351, −2.93152436547058795537535643813, −1.13669433570926874417200292648, 1.10394439480445937063231297146, 2.66298570994618849031689322357, 3.74972484114011960792590162896, 4.05811948867405575212722592240, 5.23085686472663372080321641008, 6.79854633678343318093428196368, 7.64983896144914835888491255658, 8.570155909747956475468585059651, 8.904379337730128648622381995600, 10.10848433320745967875286667970

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