Properties

Label 2-966-161.160-c1-0-30
Degree $2$
Conductor $966$
Sign $-0.533 + 0.845i$
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  + 2-s i·3-s + 4-s − 0.952·5-s i·6-s + (−0.691 − 2.55i)7-s + 8-s − 9-s − 0.952·10-s + 1.02i·11-s i·12-s − 5.28i·13-s + (−0.691 − 2.55i)14-s + 0.952i·15-s + 16-s − 1.38·17-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577i·3-s + 0.5·4-s − 0.426·5-s − 0.408i·6-s + (−0.261 − 0.965i)7-s + 0.353·8-s − 0.333·9-s − 0.301·10-s + 0.308i·11-s − 0.288i·12-s − 1.46i·13-s + (−0.184 − 0.682i)14-s + 0.246i·15-s + 0.250·16-s − 0.335·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.533 + 0.845i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.533 + 0.845i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(966\)    =    \(2 \cdot 3 \cdot 7 \cdot 23\)
Sign: $-0.533 + 0.845i$
Analytic conductor: \(7.71354\)
Root analytic conductor: \(2.77732\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{966} (643, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 966,\ (\ :1/2),\ -0.533 + 0.845i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.837013 - 1.51841i\)
\(L(\frac12)\) \(\approx\) \(0.837013 - 1.51841i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 + iT \)
7 \( 1 + (0.691 + 2.55i)T \)
23 \( 1 + (-3.53 + 3.24i)T \)
good5 \( 1 + 0.952T + 5T^{2} \)
11 \( 1 - 1.02iT - 11T^{2} \)
13 \( 1 + 5.28iT - 13T^{2} \)
17 \( 1 + 1.38T + 17T^{2} \)
19 \( 1 + 5.99T + 19T^{2} \)
29 \( 1 - 5.57T + 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 - 0.429iT - 37T^{2} \)
41 \( 1 + 1.28iT - 41T^{2} \)
43 \( 1 + 4.67iT - 43T^{2} \)
47 \( 1 + 7.09iT - 47T^{2} \)
53 \( 1 + 8.75iT - 53T^{2} \)
59 \( 1 + 1.70iT - 59T^{2} \)
61 \( 1 - 7.37T + 61T^{2} \)
67 \( 1 - 11.6iT - 67T^{2} \)
71 \( 1 + 2.29T + 71T^{2} \)
73 \( 1 - 2.19iT - 73T^{2} \)
79 \( 1 - 9.77iT - 79T^{2} \)
83 \( 1 - 5.46T + 83T^{2} \)
89 \( 1 - 8.69T + 89T^{2} \)
97 \( 1 - 19.2T + 97T^{2} \)
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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

−10.11221713764992965422749928822, −8.591501010012835094102434661248, −7.932378791856798901724681144784, −7.02951225692076667394319511343, −6.45969615381602360054655383192, −5.33717984809618627898059569658, −4.32631017755209057997648510398, −3.43698011746158899696201732281, −2.29288062733223649849704586312, −0.61115702707839104302933696082, 2.04674014584405730668731531213, 3.14307496943043744485968015770, 4.19823992197028465995025598783, 4.84847559334398801832761405267, 6.04286538199677056282294346119, 6.56449066978545468283891816866, 7.79757518962227863159342605610, 8.823749098729393859413097839222, 9.322153985296398681556716898921, 10.44661623467180322967260605558

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