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Properties

Label 2-966-69.68-c1-0-8
Degree $2$
Conductor $966$
Sign $0.879 - 0.475i$
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 + (−1.33 − 1.10i)3-s − 4-s + 0.136·5-s + (−1.10 + 1.33i)6-s − i·7-s + i·8-s + (0.574 + 2.94i)9-s − 0.136i·10-s − 1.44·11-s + (1.33 + 1.10i)12-s + 0.583·13-s − 14-s + (−0.182 − 0.150i)15-s + 16-s − 4.60·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (−0.771 − 0.635i)3-s − 0.5·4-s + 0.0612·5-s + (−0.449 + 0.545i)6-s − 0.377i·7-s + 0.353i·8-s + (0.191 + 0.981i)9-s − 0.0432i·10-s − 0.435·11-s + (0.385 + 0.317i)12-s + 0.161·13-s − 0.267·14-s + (−0.0472 − 0.0389i)15-s + 0.250·16-s − 1.11·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.879 - 0.475i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.879 - 0.475i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(966\)    =    \(2 \cdot 3 \cdot 7 \cdot 23\)
Sign: $0.879 - 0.475i$
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: \((2,\ 966,\ (\ :1/2),\ 0.879 - 0.475i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.593614 + 0.150035i\)
\(L(\frac12)\) \(\approx\) \(0.593614 + 0.150035i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.33 + 1.10i)T \)
7 \( 1 + iT \)
23 \( 1 + (1.80 - 4.44i)T \)
good5 \( 1 - 0.136T + 5T^{2} \)
11 \( 1 + 1.44T + 11T^{2} \)
13 \( 1 - 0.583T + 13T^{2} \)
17 \( 1 + 4.60T + 17T^{2} \)
19 \( 1 - 7.52iT - 19T^{2} \)
29 \( 1 + 1.43iT - 29T^{2} \)
31 \( 1 - 8.86T + 31T^{2} \)
37 \( 1 - 2.06iT - 37T^{2} \)
41 \( 1 + 4.66iT - 41T^{2} \)
43 \( 1 - 11.8iT - 43T^{2} \)
47 \( 1 - 11.3iT - 47T^{2} \)
53 \( 1 - 7.34T + 53T^{2} \)
59 \( 1 + 8.60iT - 59T^{2} \)
61 \( 1 + 12.4iT - 61T^{2} \)
67 \( 1 - 11.2iT - 67T^{2} \)
71 \( 1 + 3.92iT - 71T^{2} \)
73 \( 1 - 10.7T + 73T^{2} \)
79 \( 1 - 4.84iT - 79T^{2} \)
83 \( 1 + 16.0T + 83T^{2} \)
89 \( 1 - 8.79T + 89T^{2} \)
97 \( 1 - 11.0iT - 97T^{2} \)
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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

−10.19496992908619035089017457857, −9.602353304469340652378632886223, −8.167779637283970114968420630211, −7.78595916615506063894341000779, −6.51012657057277177036832773498, −5.82493129411668509053191116755, −4.76499494579501076076531443445, −3.81870809234688419064863894909, −2.36745653555051735222065212008, −1.29229051858684129361014707548, 0.33649330396512768133211859783, 2.55232270022940157659687923722, 4.06042298965890094454563560698, 4.80832620126896188432572683259, 5.62447520863127681439263568474, 6.49798892740280066563614344401, 7.13830699627661646527024428181, 8.505080088462391459421239367840, 8.975145912621250319787598314436, 9.981585000427951137297519240978

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