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Properties

Label 2-966-69.68-c1-0-44
Degree $2$
Conductor $966$
Sign $-0.996 - 0.0831i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 - 0.0831i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.996 - 0.0831i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(966\)    =    \(2 \cdot 3 \cdot 7 \cdot 23\)
Sign: $-0.996 - 0.0831i$
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: \((2,\ 966,\ (\ :1/2),\ -0.996 - 0.0831i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0528748 + 1.27026i\)
\(L(\frac12)\) \(\approx\) \(0.0528748 + 1.27026i\)
\(L(\frac{3}{2})\) 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

−9.509138278105462014948772652840, −8.705418842001629115981012693964, −8.052302381418632466313304954811, −7.28151196105638241624239175376, −6.24436858389096126505228321724, −5.04901525475137394919293870491, −3.87103284282545239153913297915, −3.01214508914486771642507276481, −2.00365794450986192608402019362, −0.52327513940215039315776478819, 2.08039714920297047191528735285, 3.51939273852841819589453060530, 4.09152129243907149051837208944, 5.53547034780743487709418121842, 5.79007657780332110678715871852, 7.41117005824291425427544642118, 8.122366209282428167550624850931, 8.452900554342373399117961910356, 9.582034272952610081536645019114, 10.22830485080677012551901653022

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