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Properties

Label 2-756-28.3-c1-0-51
Degree $2$
Conductor $756$
Sign $0.649 + 0.759i$
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  + (1.31 − 0.518i)2-s + (1.46 − 1.36i)4-s + (2.03 − 1.17i)5-s + (2.03 + 1.69i)7-s + (1.21 − 2.55i)8-s + (2.06 − 2.59i)10-s + (−2.18 − 1.26i)11-s + 1.48i·13-s + (3.55 + 1.17i)14-s + (0.274 − 3.99i)16-s + (1.66 + 0.958i)17-s + (−0.454 − 0.786i)19-s + (1.36 − 4.48i)20-s + (−3.53 − 0.526i)22-s + (−4.55 + 2.63i)23-s + ⋯
L(s)  = 1  + (0.930 − 0.366i)2-s + (0.730 − 0.682i)4-s + (0.908 − 0.524i)5-s + (0.768 + 0.639i)7-s + (0.429 − 0.902i)8-s + (0.652 − 0.821i)10-s + (−0.659 − 0.380i)11-s + 0.411i·13-s + (0.949 + 0.312i)14-s + (0.0687 − 0.997i)16-s + (0.402 + 0.232i)17-s + (−0.104 − 0.180i)19-s + (0.306 − 1.00i)20-s + (−0.753 − 0.112i)22-s + (−0.950 + 0.548i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.649 + 0.759i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.649 + 0.759i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $0.649 + 0.759i$
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: \((2,\ 756,\ (\ :1/2),\ 0.649 + 0.759i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.93365 - 1.35122i\)
\(L(\frac12)\) \(\approx\) \(2.93365 - 1.35122i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.31 + 0.518i)T \)
3 \( 1 \)
7 \( 1 + (-2.03 - 1.69i)T \)
good5 \( 1 + (-2.03 + 1.17i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (2.18 + 1.26i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 1.48iT - 13T^{2} \)
17 \( 1 + (-1.66 - 0.958i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.454 + 0.786i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (4.55 - 2.63i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 4.85T + 29T^{2} \)
31 \( 1 + (3.77 - 6.53i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.63 + 8.03i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 12.0iT - 41T^{2} \)
43 \( 1 - 10.9iT - 43T^{2} \)
47 \( 1 + (2.04 + 3.54i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.83 - 4.91i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.98 + 6.90i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.72 + 3.30i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.36 + 0.790i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 0.670iT - 71T^{2} \)
73 \( 1 + (-2.49 - 1.44i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (13.5 - 7.81i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 6.82T + 83T^{2} \)
89 \( 1 + (-3.23 + 1.86i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 14.0iT - 97T^{2} \)
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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

−10.41212096419764907835631040944, −9.491379306351214328893029118188, −8.632036810694177343373974591788, −7.56115155555681757268284253495, −6.33880340141263588034984605722, −5.44534948886432367312116728336, −5.07644127046583940941520937534, −3.76271968267499229149568858800, −2.40495883439777950738762246981, −1.54089689763529319272849295872, 1.88615000324644046750093233227, 2.91056373188688390274417916548, 4.20596427319332675624224647380, 5.11275527173969075183449625933, 5.96207063825652413129806741832, 6.83092342334353479600706399930, 7.73698265951537758376126444331, 8.384325986610858285256612764109, 10.04267811981243405519549804431, 10.35970237074315533529777313361

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