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Properties

Label 2-966-69.68-c1-0-31
Degree $2$
Conductor $966$
Sign $0.0741 + 0.997i$
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 + (1.23 − 1.21i)3-s − 4-s + 0.666·5-s + (−1.21 − 1.23i)6-s + i·7-s + i·8-s + (0.0269 − 2.99i)9-s − 0.666i·10-s + 5.34·11-s + (−1.23 + 1.21i)12-s + 5.36·13-s + 14-s + (0.820 − 0.812i)15-s + 16-s + 0.110·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (0.710 − 0.703i)3-s − 0.5·4-s + 0.298·5-s + (−0.497 − 0.502i)6-s + 0.377i·7-s + 0.353i·8-s + (0.00899 − 0.999i)9-s − 0.210i·10-s + 1.61·11-s + (−0.355 + 0.351i)12-s + 1.48·13-s + 0.267·14-s + (0.211 − 0.209i)15-s + 0.250·16-s + 0.0267·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0741 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0741 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(966\)    =    \(2 \cdot 3 \cdot 7 \cdot 23\)
Sign: $0.0741 + 0.997i$
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: \((2,\ 966,\ (\ :1/2),\ 0.0741 + 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.67796 - 1.55784i\)
\(L(\frac12)\) \(\approx\) \(1.67796 - 1.55784i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.23 + 1.21i)T \)
7 \( 1 - iT \)
23 \( 1 + (-3.11 + 3.64i)T \)
good5 \( 1 - 0.666T + 5T^{2} \)
11 \( 1 - 5.34T + 11T^{2} \)
13 \( 1 - 5.36T + 13T^{2} \)
17 \( 1 - 0.110T + 17T^{2} \)
19 \( 1 - 7.89iT - 19T^{2} \)
29 \( 1 - 5.21iT - 29T^{2} \)
31 \( 1 + 6.24T + 31T^{2} \)
37 \( 1 + 4.98iT - 37T^{2} \)
41 \( 1 + 10.3iT - 41T^{2} \)
43 \( 1 + 7.96iT - 43T^{2} \)
47 \( 1 - 4.18iT - 47T^{2} \)
53 \( 1 + 9.76T + 53T^{2} \)
59 \( 1 + 2.11iT - 59T^{2} \)
61 \( 1 - 3.58iT - 61T^{2} \)
67 \( 1 - 8.90iT - 67T^{2} \)
71 \( 1 + 6.81iT - 71T^{2} \)
73 \( 1 - 9.22T + 73T^{2} \)
79 \( 1 - 5.18iT - 79T^{2} \)
83 \( 1 + 9.21T + 83T^{2} \)
89 \( 1 + 3.34T + 89T^{2} \)
97 \( 1 + 1.57iT - 97T^{2} \)
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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

−9.651041891305277912322768158088, −8.888246329188101027964824471839, −8.544291901237605887916848992649, −7.36849008613898893650383926714, −6.33408509612949808739229865205, −5.68515137220746321967720017820, −3.87693205909693512789871745788, −3.54556233650152540880243918681, −1.99790671870339390868305321036, −1.28894698638607008974857718019, 1.44730362707573117522110564356, 3.20863993971522430542736842261, 4.02585268059015827024179240318, 4.84658793913425946936026617418, 6.06258060606172973839151104180, 6.77191210838749363455040446511, 7.80550830018740960842994399309, 8.656606700685744517737108709050, 9.354718163055611894780494595871, 9.747242234037169014038160460815

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