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Properties

Label 2-756-28.19-c1-0-39
Degree $2$
Conductor $756$
Sign $0.997 - 0.0686i$
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  + (0.434 + 1.34i)2-s + (−1.62 + 1.16i)4-s + (−0.421 − 0.243i)5-s + (−0.250 − 2.63i)7-s + (−2.27 − 1.67i)8-s + (0.144 − 0.673i)10-s + (2.51 − 1.45i)11-s − 1.17i·13-s + (3.43 − 1.48i)14-s + (1.26 − 3.79i)16-s + (−0.0401 + 0.0231i)17-s + (2.38 − 4.12i)19-s + (0.969 − 0.0978i)20-s + (3.04 + 2.75i)22-s + (6.02 + 3.47i)23-s + ⋯
L(s)  = 1  + (0.307 + 0.951i)2-s + (−0.811 + 0.584i)4-s + (−0.188 − 0.108i)5-s + (−0.0946 − 0.995i)7-s + (−0.805 − 0.592i)8-s + (0.0457 − 0.212i)10-s + (0.757 − 0.437i)11-s − 0.325i·13-s + (0.918 − 0.395i)14-s + (0.316 − 0.948i)16-s + (−0.00973 + 0.00561i)17-s + (0.546 − 0.946i)19-s + (0.216 − 0.0218i)20-s + (0.648 + 0.586i)22-s + (1.25 + 0.725i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0686i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 - 0.0686i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $0.997 - 0.0686i$
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: \((2,\ 756,\ (\ :1/2),\ 0.997 - 0.0686i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.47910 + 0.0508037i\)
\(L(\frac12)\) \(\approx\) \(1.47910 + 0.0508037i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.434 - 1.34i)T \)
3 \( 1 \)
7 \( 1 + (0.250 + 2.63i)T \)
good5 \( 1 + (0.421 + 0.243i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (-2.51 + 1.45i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 1.17iT - 13T^{2} \)
17 \( 1 + (0.0401 - 0.0231i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.38 + 4.12i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-6.02 - 3.47i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 0.465T + 29T^{2} \)
31 \( 1 + (-0.536 - 0.928i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.196 + 0.339i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 0.139iT - 41T^{2} \)
43 \( 1 + 6.47iT - 43T^{2} \)
47 \( 1 + (-3.81 + 6.61i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-5.03 - 8.71i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.65 - 6.32i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (12.0 + 6.95i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.49 + 2.01i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 6.95iT - 71T^{2} \)
73 \( 1 + (7.14 - 4.12i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-12.0 - 6.92i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 6.34T + 83T^{2} \)
89 \( 1 + (9.16 + 5.29i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 8.51iT - 97T^{2} \)
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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

−10.23251314348113357713257812294, −9.250877051274809015678593327167, −8.576842198414714348473177863558, −7.48320540624306280943739288328, −7.00244732597735941372728815208, −6.00957856072039718428394229007, −4.96833401884896486642864420363, −4.05110805439590955175586028395, −3.15733285616532338153225364800, −0.77114508079576662188461107069, 1.46376406825697052411879879468, 2.68476709474216038898356784776, 3.71127226284285675852861046846, 4.76826795127405597436178589428, 5.70680776815202391075779892758, 6.64591715918317253130286388831, 7.952279875757150268430141051789, 9.020477307843857220991249290868, 9.444435722563262224191615172034, 10.39808151392886651766091811437

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