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Properties

Label 2-756-28.3-c1-0-28
Degree $2$
Conductor $756$
Sign $0.925 - 0.377i$
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  + (−0.145 + 1.40i)2-s + (−1.95 − 0.408i)4-s + (−3.03 + 1.75i)5-s + (0.151 − 2.64i)7-s + (0.859 − 2.69i)8-s + (−2.02 − 4.52i)10-s + (2.81 + 1.62i)11-s − 2.17i·13-s + (3.69 + 0.596i)14-s + (3.66 + 1.60i)16-s + (−4.04 − 2.33i)17-s + (0.0375 + 0.0650i)19-s + (6.66 − 2.19i)20-s + (−2.70 + 3.73i)22-s + (2.40 − 1.38i)23-s + ⋯
L(s)  = 1  + (−0.102 + 0.994i)2-s + (−0.978 − 0.204i)4-s + (−1.35 + 0.784i)5-s + (0.0571 − 0.998i)7-s + (0.303 − 0.952i)8-s + (−0.640 − 1.43i)10-s + (0.850 + 0.490i)11-s − 0.603i·13-s + (0.987 + 0.159i)14-s + (0.916 + 0.400i)16-s + (−0.980 − 0.566i)17-s + (0.00861 + 0.0149i)19-s + (1.48 − 0.489i)20-s + (−0.575 + 0.795i)22-s + (0.501 − 0.289i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.925 - 0.377i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.925 - 0.377i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $0.925 - 0.377i$
Analytic conductor: \(6.03669\)
Root analytic conductor: \(2.45696\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{756} (703, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 756,\ (\ :1/2),\ 0.925 - 0.377i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.895959 + 0.175698i\)
\(L(\frac12)\) \(\approx\) \(0.895959 + 0.175698i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.145 - 1.40i)T \)
3 \( 1 \)
7 \( 1 + (-0.151 + 2.64i)T \)
good5 \( 1 + (3.03 - 1.75i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (-2.81 - 1.62i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 2.17iT - 13T^{2} \)
17 \( 1 + (4.04 + 2.33i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.0375 - 0.0650i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.40 + 1.38i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 8.09T + 29T^{2} \)
31 \( 1 + (-3.66 + 6.34i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-5.08 - 8.80i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 7.20iT - 41T^{2} \)
43 \( 1 + 1.49iT - 43T^{2} \)
47 \( 1 + (-0.225 - 0.391i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.69 + 2.93i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-6.23 + 10.7i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-12.7 + 7.33i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (8.05 + 4.65i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 11.8iT - 71T^{2} \)
73 \( 1 + (0.852 + 0.492i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-3.09 + 1.78i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 0.350T + 83T^{2} \)
89 \( 1 + (-3.06 + 1.76i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 1.05iT - 97T^{2} \)
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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

−10.29350336820702998722268562132, −9.506482375284577132324675770439, −8.236939813151275757294640171455, −7.82544862496846538125660946016, −6.68956693663390061172651276208, −6.66058144731209880762536148934, −4.74269648653831185856962586867, −4.22263770658535568107879170072, −3.16591885242767977053741349613, −0.63997196451521149240639100605, 1.07709005202511705109188253704, 2.59684892707782489203056223637, 3.88640569213942424313785479639, 4.45814667008253045347137965782, 5.55164189009296832242454612837, 6.90612150550692835186601028925, 8.211175570966675900776985560798, 8.780357604904508702593653957253, 9.126825016879352861825234592563, 10.48056457687231736421673728611

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