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Properties

Label 2-756-28.19-c1-0-20
Degree $2$
Conductor $756$
Sign $-0.106 - 0.994i$
Analytic cond. $6.03669$
Root an. cond. $2.45696$
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  + (1.28 + 0.582i)2-s + (1.32 + 1.50i)4-s + (−1.99 − 1.15i)5-s + (0.0928 + 2.64i)7-s + (0.829 + 2.70i)8-s + (−1.90 − 2.64i)10-s + (2.38 − 1.37i)11-s + 5.36i·13-s + (−1.42 + 3.46i)14-s + (−0.506 + 3.96i)16-s + (−1.60 + 0.927i)17-s + (−1.03 + 1.78i)19-s + (−0.909 − 4.52i)20-s + (3.86 − 0.384i)22-s + (3.19 + 1.84i)23-s + ⋯
L(s)  = 1  + (0.911 + 0.411i)2-s + (0.660 + 0.750i)4-s + (−0.893 − 0.515i)5-s + (0.0350 + 0.999i)7-s + (0.293 + 0.956i)8-s + (−0.601 − 0.837i)10-s + (0.717 − 0.414i)11-s + 1.48i·13-s + (−0.379 + 0.925i)14-s + (−0.126 + 0.991i)16-s + (−0.389 + 0.224i)17-s + (−0.236 + 0.409i)19-s + (−0.203 − 1.01i)20-s + (0.824 − 0.0820i)22-s + (0.666 + 0.384i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.106 - 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.106 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $-0.106 - 0.994i$
Analytic conductor: \(6.03669\)
Root analytic conductor: \(2.45696\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{756} (271, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 756,\ (\ :1/2),\ -0.106 - 0.994i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.48121 + 1.64833i\)
\(L(\frac12)\) \(\approx\) \(1.48121 + 1.64833i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.28 - 0.582i)T \)
3 \( 1 \)
7 \( 1 + (-0.0928 - 2.64i)T \)
good5 \( 1 + (1.99 + 1.15i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (-2.38 + 1.37i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 5.36iT - 13T^{2} \)
17 \( 1 + (1.60 - 0.927i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.03 - 1.78i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.19 - 1.84i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 6.56T + 29T^{2} \)
31 \( 1 + (1.38 + 2.39i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.63 - 4.55i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 8.65iT - 41T^{2} \)
43 \( 1 + 3.57iT - 43T^{2} \)
47 \( 1 + (-2.54 + 4.41i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.523 + 0.906i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (6.66 + 11.5i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-7.62 - 4.40i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-13.7 + 7.94i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 14.2iT - 71T^{2} \)
73 \( 1 + (-5.96 + 3.44i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3.51 - 2.02i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 0.863T + 83T^{2} \)
89 \( 1 + (13.9 + 8.04i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 4.58iT - 97T^{2} \)
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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

−11.03777843747358480497122986084, −9.451096010025888022889132654386, −8.600755062562377203495503039116, −8.084978290493349901413640858584, −6.79919900215685738089526190531, −6.24395194854792443875885720514, −5.02672391213830545597047074169, −4.29195039152354281556407475093, −3.35195434152384573380927508276, −1.92399207159908803314267582376, 0.880258190601200313149182484288, 2.70815792536391066869497985852, 3.67396980888031383757109756314, 4.39322725943977103999585674526, 5.44523251454522203758096280006, 6.80602870452589404122198278780, 7.14561991848210402421994795991, 8.212153171781805597149783147431, 9.540338516385895976023496555360, 10.65087905908952912107281922530

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