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Properties

Label 2-966-69.68-c1-0-25
Degree $2$
Conductor $966$
Sign $-0.778 + 0.628i$
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 + (−1.69 − 0.349i)3-s − 4-s − 1.62·5-s + (−0.349 + 1.69i)6-s − i·7-s + i·8-s + (2.75 + 1.18i)9-s + 1.62i·10-s + 3.65·11-s + (1.69 + 0.349i)12-s + 0.597·13-s − 14-s + (2.75 + 0.567i)15-s + 16-s + 3.33·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (−0.979 − 0.201i)3-s − 0.5·4-s − 0.726·5-s + (−0.142 + 0.692i)6-s − 0.377i·7-s + 0.353i·8-s + (0.918 + 0.394i)9-s + 0.513i·10-s + 1.10·11-s + (0.489 + 0.100i)12-s + 0.165·13-s − 0.267·14-s + (0.711 + 0.146i)15-s + 0.250·16-s + 0.808·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.778 + 0.628i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.778 + 0.628i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(966\)    =    \(2 \cdot 3 \cdot 7 \cdot 23\)
Sign: $-0.778 + 0.628i$
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: \((2,\ 966,\ (\ :1/2),\ -0.778 + 0.628i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.262096 - 0.741889i\)
\(L(\frac12)\) \(\approx\) \(0.262096 - 0.741889i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.69 + 0.349i)T \)
7 \( 1 + iT \)
23 \( 1 + (-3.04 + 3.70i)T \)
good5 \( 1 + 1.62T + 5T^{2} \)
11 \( 1 - 3.65T + 11T^{2} \)
13 \( 1 - 0.597T + 13T^{2} \)
17 \( 1 - 3.33T + 17T^{2} \)
19 \( 1 - 0.263iT - 19T^{2} \)
29 \( 1 + 1.04iT - 29T^{2} \)
31 \( 1 + 8.47T + 31T^{2} \)
37 \( 1 + 7.11iT - 37T^{2} \)
41 \( 1 - 5.26iT - 41T^{2} \)
43 \( 1 - 0.223iT - 43T^{2} \)
47 \( 1 + 5.31iT - 47T^{2} \)
53 \( 1 - 4.55T + 53T^{2} \)
59 \( 1 + 6.54iT - 59T^{2} \)
61 \( 1 + 8.08iT - 61T^{2} \)
67 \( 1 + 12.2iT - 67T^{2} \)
71 \( 1 - 3.11iT - 71T^{2} \)
73 \( 1 + 7.94T + 73T^{2} \)
79 \( 1 + 8.69iT - 79T^{2} \)
83 \( 1 - 8.43T + 83T^{2} \)
89 \( 1 + 7.18T + 89T^{2} \)
97 \( 1 - 11.3iT - 97T^{2} \)
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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

−9.880288598451810109467024834612, −9.059507461567687638302148707966, −7.930804728722125718877293116150, −7.16350072274235562235564646520, −6.23050617724559840375864519530, −5.20235314881407703247873826082, −4.19098771314825179634282587517, −3.52194458918872831410956411965, −1.73403010224411777490408241292, −0.50005022122781913922312489526, 1.24019666798117283974806226710, 3.50767572652649839183066181344, 4.25355233732739808820751671727, 5.34716869098911745469055228403, 5.97579946154366746734088691859, 6.99266759563561141084317830998, 7.54590598768240429924784994070, 8.715450446779324445911633615007, 9.415275716565240758829216956412, 10.29912387517678797284817273227

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