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Properties

Label 2-966-69.68-c1-0-3
Degree $2$
Conductor $966$
Sign $-0.406 + 0.913i$
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 + (−0.0866 + 1.72i)3-s − 4-s + 0.713·5-s + (−1.72 − 0.0866i)6-s − i·7-s − i·8-s + (−2.98 − 0.299i)9-s + 0.713i·10-s + 2.60·11-s + (0.0866 − 1.72i)12-s − 5.59·13-s + 14-s + (−0.0618 + 1.23i)15-s + 16-s − 6.17·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (−0.0500 + 0.998i)3-s − 0.5·4-s + 0.319·5-s + (−0.706 − 0.0353i)6-s − 0.377i·7-s − 0.353i·8-s + (−0.994 − 0.0999i)9-s + 0.225i·10-s + 0.784·11-s + (0.0250 − 0.499i)12-s − 1.55·13-s + 0.267·14-s + (−0.0159 + 0.318i)15-s + 0.250·16-s − 1.49·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.406 + 0.913i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.406 + 0.913i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(966\)    =    \(2 \cdot 3 \cdot 7 \cdot 23\)
Sign: $-0.406 + 0.913i$
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: \((2,\ 966,\ (\ :1/2),\ -0.406 + 0.913i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.177294 - 0.272805i\)
\(L(\frac12)\) \(\approx\) \(0.177294 - 0.272805i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.0866 - 1.72i)T \)
7 \( 1 + iT \)
23 \( 1 + (4.47 + 1.72i)T \)
good5 \( 1 - 0.713T + 5T^{2} \)
11 \( 1 - 2.60T + 11T^{2} \)
13 \( 1 + 5.59T + 13T^{2} \)
17 \( 1 + 6.17T + 17T^{2} \)
19 \( 1 - 3.14iT - 19T^{2} \)
29 \( 1 + 0.797iT - 29T^{2} \)
31 \( 1 + 7.70T + 31T^{2} \)
37 \( 1 - 7.21iT - 37T^{2} \)
41 \( 1 - 3.84iT - 41T^{2} \)
43 \( 1 + 4.89iT - 43T^{2} \)
47 \( 1 - 4.60iT - 47T^{2} \)
53 \( 1 - 13.4T + 53T^{2} \)
59 \( 1 + 11.2iT - 59T^{2} \)
61 \( 1 + 1.01iT - 61T^{2} \)
67 \( 1 - 13.9iT - 67T^{2} \)
71 \( 1 - 4.29iT - 71T^{2} \)
73 \( 1 - 7.05T + 73T^{2} \)
79 \( 1 - 5.48iT - 79T^{2} \)
83 \( 1 + 10.1T + 83T^{2} \)
89 \( 1 - 8.77T + 89T^{2} \)
97 \( 1 + 14.9iT - 97T^{2} \)
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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

−10.20045347153238927489516227357, −9.778270707883221242898823497215, −9.007538466168428200159020788840, −8.164372306348272000469281616096, −7.12946438208521769756625272119, −6.27881098704227105457304617429, −5.37778001561949787434265975993, −4.45121936588826078120408466639, −3.79959776369912966132938176462, −2.24741601123925441156094171887, 0.13859436241548852735905986705, 1.94092938303859418236693420823, 2.43011282101633326699583594027, 3.90181989120746697934054486731, 5.09101172033551592553317806697, 5.99238967700121013516058006842, 6.98217041567362105121109603080, 7.68540530807073234950523940464, 8.962134966242794527020334447254, 9.223811947990099079369107700117

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