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Properties

Label 2-756-21.11-c2-0-10
Degree $2$
Conductor $756$
Sign $0.756 - 0.654i$
Analytic cond. $20.5995$
Root an. cond. $4.53866$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.181 + 0.104i)5-s + (6.74 + 1.87i)7-s + (16.4 + 9.50i)11-s − 7.60·13-s + (4.04 + 2.33i)17-s + (−9.11 − 15.7i)19-s + (3.20 − 1.84i)23-s + (−12.4 + 21.6i)25-s + 25.8i·29-s + (4.91 − 8.51i)31-s + (−1.42 + 0.367i)35-s + (−3.12 − 5.41i)37-s + 17.8i·41-s + 41.0·43-s + (25.7 − 14.8i)47-s + ⋯
L(s)  = 1  + (−0.0363 + 0.0209i)5-s + (0.963 + 0.267i)7-s + (1.49 + 0.864i)11-s − 0.585·13-s + (0.238 + 0.137i)17-s + (−0.479 − 0.830i)19-s + (0.139 − 0.0803i)23-s + (−0.499 + 0.864i)25-s + 0.892i·29-s + (0.158 − 0.274i)31-s + (−0.0406 + 0.0105i)35-s + (−0.0845 − 0.146i)37-s + 0.435i·41-s + 0.954·43-s + (0.546 − 0.315i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.756 - 0.654i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.756 - 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $0.756 - 0.654i$
Analytic conductor: \(20.5995\)
Root analytic conductor: \(4.53866\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{756} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 756,\ (\ :1),\ 0.756 - 0.654i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.135470648\)
\(L(\frac12)\) \(\approx\) \(2.135470648\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 + (-6.74 - 1.87i)T \)
good5 \( 1 + (0.181 - 0.104i)T + (12.5 - 21.6i)T^{2} \)
11 \( 1 + (-16.4 - 9.50i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + 7.60T + 169T^{2} \)
17 \( 1 + (-4.04 - 2.33i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (9.11 + 15.7i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (-3.20 + 1.84i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 - 25.8iT - 841T^{2} \)
31 \( 1 + (-4.91 + 8.51i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + (3.12 + 5.41i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 - 17.8iT - 1.68e3T^{2} \)
43 \( 1 - 41.0T + 1.84e3T^{2} \)
47 \( 1 + (-25.7 + 14.8i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-69.4 - 40.1i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (-69.3 - 40.0i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (29.2 + 50.7i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (45.4 - 78.6i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 88.2iT - 5.04e3T^{2} \)
73 \( 1 + (-10.0 + 17.3i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (32.1 + 55.7i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 - 37.3iT - 6.88e3T^{2} \)
89 \( 1 + (-123. + 71.3i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + 44.6T + 9.40e3T^{2} \)
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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

−10.22507074880427647920949382079, −9.237810095643651222737591280349, −8.718536487603193074923092798030, −7.50157032161673289380913208036, −6.93292882196762022696280340216, −5.73891313516503428836747268513, −4.72858868491177196289777114781, −3.94801011158224745951532441430, −2.39566202323226050790795072086, −1.28697632059858695755585637190, 0.862472950988414583934914639976, 2.13531783199113051056279786921, 3.69567648351587349781253648555, 4.44273164716813130117793826495, 5.64244474050188592966829885465, 6.49114904816365062698220130542, 7.55355613535729194517319802044, 8.336799432563944799488200957005, 9.106871312453372781146395835961, 10.09884580203940557932561439378

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