Properties

Degree $2$
Conductor $1764$
Sign $-0.224 + 0.974i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.629 + 1.26i)2-s + (−1.20 + 1.59i)4-s − 2.32i·5-s + (−2.77 − 0.524i)8-s + (2.94 − 1.46i)10-s + 3.58i·11-s − 2.93i·13-s + (−1.08 − 3.84i)16-s − 2.32i·17-s − 8.33·19-s + (3.71 + 2.80i)20-s + (−4.53 + 2.25i)22-s + 1.48i·23-s − 0.414·25-s + (3.71 − 1.84i)26-s + ⋯
L(s)  = 1  + (0.445 + 0.895i)2-s + (−0.603 + 0.797i)4-s − 1.04i·5-s + (−0.982 − 0.185i)8-s + (0.931 − 0.463i)10-s + 1.07i·11-s − 0.812i·13-s + (−0.271 − 0.962i)16-s − 0.564i·17-s − 1.91·19-s + (0.829 + 0.628i)20-s + (−0.966 + 0.480i)22-s + 0.309i·23-s − 0.0828·25-s + (0.727 − 0.361i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.224 + 0.974i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.224 + 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.224 + 0.974i$
Motivic weight: \(1\)
Character: $\chi_{1764} (1567, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1764,\ (\ :1/2),\ -0.224 + 0.974i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4082443653\)
\(L(\frac12)\) \(\approx\) \(0.4082443653\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.629 - 1.26i)T \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + 2.32iT - 5T^{2} \)
11 \( 1 - 3.58iT - 11T^{2} \)
13 \( 1 + 2.93iT - 13T^{2} \)
17 \( 1 + 2.32iT - 17T^{2} \)
19 \( 1 + 8.33T + 19T^{2} \)
23 \( 1 - 1.48iT - 23T^{2} \)
29 \( 1 + 7.86T + 29T^{2} \)
31 \( 1 - 3.45T + 31T^{2} \)
37 \( 1 + 8.24T + 37T^{2} \)
41 \( 1 - 2.89iT - 41T^{2} \)
43 \( 1 + 6.37iT - 43T^{2} \)
47 \( 1 + 10.4T + 47T^{2} \)
53 \( 1 + 8.59T + 53T^{2} \)
59 \( 1 - 10.4T + 59T^{2} \)
61 \( 1 - 2.93iT - 61T^{2} \)
67 \( 1 + 9.02iT - 67T^{2} \)
71 \( 1 - 10.7iT - 71T^{2} \)
73 \( 1 + 11.2iT - 73T^{2} \)
79 \( 1 + 15.3iT - 79T^{2} \)
83 \( 1 + 14.8T + 83T^{2} \)
89 \( 1 + 13.5iT - 89T^{2} \)
97 \( 1 + 5.35iT - 97T^{2} \)
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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

−8.828913197543262106225961793788, −8.276226305101177095761792810540, −7.44736124022621929320759528481, −6.69528748430994335583095451804, −5.74448875464127994912834489274, −4.93022500417526827088284054695, −4.43657534319352407734522264689, −3.36508178558432113262837476947, −1.94626271765675220942288039642, −0.12285932077456366592653348894, 1.71027101250864604033837361318, 2.65304858274090204557959378697, 3.58436698889537040021688145056, 4.26692688575319013095584923134, 5.42870758181234802767870704713, 6.37238460163971912622400895192, 6.75422201029676381678164145769, 8.235274828103559691629058640727, 8.777941902005951687554046905889, 9.763690888125158488866451633390

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