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Properties

Label 2-756-21.2-c2-0-6
Degree $2$
Conductor $756$
Sign $0.997 - 0.0633i$
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  + (−5.19 − 3i)5-s + (−3.5 + 6.06i)7-s + (2.59 − 1.5i)11-s − 22·13-s + (20.7 − 12i)17-s + (5 − 8.66i)19-s + (25.9 + 15i)23-s + (5.5 + 9.52i)25-s + 33i·29-s + (6.5 + 11.2i)31-s + (36.3 − 21i)35-s + (5 − 8.66i)37-s + 30i·41-s + 56·43-s + (−24.5 − 42.4i)49-s + ⋯
L(s)  = 1  + (−1.03 − 0.600i)5-s + (−0.5 + 0.866i)7-s + (0.236 − 0.136i)11-s − 1.69·13-s + (1.22 − 0.705i)17-s + (0.263 − 0.455i)19-s + (1.12 + 0.652i)23-s + (0.220 + 0.381i)25-s + 1.13i·29-s + (0.209 + 0.363i)31-s + (1.03 − 0.599i)35-s + (0.135 − 0.234i)37-s + 0.731i·41-s + 1.30·43-s + (−0.499 − 0.866i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0633i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.997 - 0.0633i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $0.997 - 0.0633i$
Analytic conductor: \(20.5995\)
Root analytic conductor: \(4.53866\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{756} (485, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 756,\ (\ :1),\ 0.997 - 0.0633i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.197292244\)
\(L(\frac12)\) \(\approx\) \(1.197292244\)
\(L(2)\) 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 \( 1 + (3.5 - 6.06i)T \)
good5 \( 1 + (5.19 + 3i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (-2.59 + 1.5i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + 22T + 169T^{2} \)
17 \( 1 + (-20.7 + 12i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (-5 + 8.66i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (-25.9 - 15i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 - 33iT - 841T^{2} \)
31 \( 1 + (-6.5 - 11.2i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + (-5 + 8.66i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 - 30iT - 1.68e3T^{2} \)
43 \( 1 - 56T + 1.84e3T^{2} \)
47 \( 1 + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-36.3 + 21i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-85.7 + 49.5i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-47 + 81.4i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-38 - 65.8i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 18iT - 5.04e3T^{2} \)
73 \( 1 + (-27.5 - 47.6i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (41.5 - 71.8i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + 147iT - 6.88e3T^{2} \)
89 \( 1 + (-114. - 66i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + 85T + 9.40e3T^{2} \)
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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

−9.883899373472441773791273732821, −9.336074561743640795192177918988, −8.472827589754962314484725509335, −7.53154011161439237999885817397, −6.88523595533162597172261581702, −5.38283168839889902671806442852, −4.91735512113083438358805941135, −3.54978316396034327039593527801, −2.61541663069903229026800161708, −0.74433770896121186224523746675, 0.68061876501599533113599740219, 2.63200278302785379151226611067, 3.67663304632969522971428345637, 4.42976791877445013147520523224, 5.72634794472314688338524472360, 6.99259339785824013411827380047, 7.39830054545161679094720566936, 8.159109823589827077306920948054, 9.512722188135337652529525090064, 10.16469635456216194151185057552

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