↑

Properties

Label 2-1452-11.4-c1-0-3
Degree $2$
Conductor $1452$
Sign $0.220 - 0.975i$
Analytic cond. $11.5942$
Root an. cond. $3.40503$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.309 − 0.951i)3-s + (−1.61 + 1.17i)5-s + (−0.618 − 1.90i)7-s + (−0.809 − 0.587i)9-s + (1.61 + 1.17i)13-s + (0.618 + 1.90i)15-s + (−3.23 + 2.35i)17-s + (−1.85 + 5.70i)19-s − 1.99·21-s + (−0.309 + 0.951i)25-s + (−0.809 + 0.587i)27-s + (−2.47 − 7.60i)29-s + (6.47 + 4.70i)31-s + (3.23 + 2.35i)35-s + (3.09 + 9.51i)37-s + ⋯
L(s)  = 1  + (0.178 − 0.549i)3-s + (−0.723 + 0.525i)5-s + (−0.233 − 0.718i)7-s + (−0.269 − 0.195i)9-s + (0.448 + 0.326i)13-s + (0.159 + 0.491i)15-s + (−0.784 + 0.570i)17-s + (−0.425 + 1.30i)19-s − 0.436·21-s + (−0.0618 + 0.190i)25-s + (−0.155 + 0.113i)27-s + (−0.459 − 1.41i)29-s + (1.16 + 0.844i)31-s + (0.546 + 0.397i)35-s + (0.508 + 1.56i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1452 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.220 - 0.975i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1452 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.220 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1452\)    =    \(2^{2} \cdot 3 \cdot 11^{2}\)
Sign: $0.220 - 0.975i$
Analytic conductor: \(11.5942\)
Root analytic conductor: \(3.40503\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1452} (565, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1452,\ (\ :1/2),\ 0.220 - 0.975i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9415301414\)
\(L(\frac12)\) \(\approx\) \(0.9415301414\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.309 + 0.951i)T \)
11 \( 1 \)
good5 \( 1 + (1.61 - 1.17i)T + (1.54 - 4.75i)T^{2} \)
7 \( 1 + (0.618 + 1.90i)T + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (-1.61 - 1.17i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (3.23 - 2.35i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (1.85 - 5.70i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 + (2.47 + 7.60i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-6.47 - 4.70i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-3.09 - 9.51i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-2.47 + 7.60i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 2T + 43T^{2} \)
47 \( 1 + (2.47 - 7.60i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-1.61 - 1.17i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-3.70 - 11.4i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (8.09 - 5.87i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 - 12T + 67T^{2} \)
71 \( 1 + (6.47 - 4.70i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-1.85 - 5.70i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-1.61 - 1.17i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (12.9 - 9.40i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + 14T + 89T^{2} \)
97 \( 1 + (-1.61 - 1.17i)T + (29.9 + 92.2i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.808974729445785901685920223589, −8.616183514240651797911460860525, −8.066405403683077160795452766893, −7.26374069005362499239193116984, −6.57144541979552213694036748466, −5.84935027549625780893554773399, −4.28577491007090910506992851342, −3.78440163196704867483761134883, −2.64092181844543944400272841859, −1.30643677708535625839746113628, 0.38884075659748479317094219263, 2.30375891111222766403500399565, 3.28020440940553587825647715232, 4.35358517783250161771576260131, 4.96380189584177718186054312955, 5.98800012339408638719577623505, 6.92930001810672060663747287420, 7.955976280972087366052590656744, 8.712804294916946340653807897288, 9.148727711552150290251981201405

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