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Properties

Label 2-756-21.11-c2-0-19
Degree $2$
Conductor $756$
Sign $-0.714 + 0.699i$
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  + (6.05 − 3.49i)5-s + (−1.44 + 6.84i)7-s + (−9.05 − 5.22i)11-s − 15.3·13-s + (−26.6 − 15.4i)17-s + (−12.8 − 22.2i)19-s + (23.6 − 13.6i)23-s + (11.9 − 20.6i)25-s − 0.0542i·29-s + (0.305 − 0.529i)31-s + (15.1 + 46.5i)35-s + (−22.4 − 38.9i)37-s + 65.8i·41-s + 14.0·43-s + (42.2 − 24.4i)47-s + ⋯
L(s)  = 1  + (1.21 − 0.698i)5-s + (−0.207 + 0.978i)7-s + (−0.823 − 0.475i)11-s − 1.17·13-s + (−1.57 − 0.906i)17-s + (−0.676 − 1.17i)19-s + (1.02 − 0.593i)23-s + (0.476 − 0.825i)25-s − 0.00186i·29-s + (0.00986 − 0.0170i)31-s + (0.432 + 1.32i)35-s + (−0.607 − 1.05i)37-s + 1.60i·41-s + 0.326·43-s + (0.899 − 0.519i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.714 + 0.699i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.714 + 0.699i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $-0.714 + 0.699i$
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: \((2,\ 756,\ (\ :1),\ -0.714 + 0.699i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.9544288344\)
\(L(\frac12)\) \(\approx\) \(0.9544288344\)
\(L(2)\) 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 \( 1 + (1.44 - 6.84i)T \)
good5 \( 1 + (-6.05 + 3.49i)T + (12.5 - 21.6i)T^{2} \)
11 \( 1 + (9.05 + 5.22i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + 15.3T + 169T^{2} \)
17 \( 1 + (26.6 + 15.4i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (12.8 + 22.2i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (-23.6 + 13.6i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + 0.0542iT - 841T^{2} \)
31 \( 1 + (-0.305 + 0.529i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + (22.4 + 38.9i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 - 65.8iT - 1.68e3T^{2} \)
43 \( 1 - 14.0T + 1.84e3T^{2} \)
47 \( 1 + (-42.2 + 24.4i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (54.9 + 31.7i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (-68.4 - 39.5i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (53.1 + 92.0i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-29.9 + 51.9i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 20.3iT - 5.04e3T^{2} \)
73 \( 1 + (32.4 - 56.2i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-16.9 - 29.3i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 - 57.0iT - 6.88e3T^{2} \)
89 \( 1 + (-129. + 74.8i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 - 56.2T + 9.40e3T^{2} \)
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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

−9.498807578017064892636097834263, −9.153316964391714013843398651871, −8.400718467815461246291430254744, −7.04206557074922127201497839198, −6.22512214980078833791344770722, −5.11905112879343808636364296144, −4.82737172774479606038100427789, −2.69483722536379765361576833292, −2.22364882792155072127221219341, −0.29095243087827926519806786470, 1.79025511018201048184342850930, 2.68806747172333698404182933140, 4.06638298366068295997253288217, 5.12285792054253401027304993282, 6.19151088192888069017130455068, 6.93985567363439732530573414147, 7.67197531032463967175371040978, 8.916255703586659936541485168163, 9.853440398919901927349217677315, 10.48852199644124550126403575188

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