↑

Properties

Label 2-648-72.11-c1-0-14
Degree $2$
Conductor $648$
Sign $0.653 - 0.756i$
Analytic cond. $5.17430$
Root an. cond. $2.27471$
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  + (0.238 + 1.39i)2-s + (−1.88 + 0.666i)4-s + (−0.796 − 1.37i)5-s + (−1.24 − 0.719i)7-s + (−1.37 − 2.46i)8-s + (1.73 − 1.43i)10-s + (4.27 + 2.46i)11-s + (4.65 − 2.68i)13-s + (0.705 − 1.90i)14-s + (3.11 − 2.51i)16-s + 1.32i·17-s + 0.267·19-s + (2.42 + 2.07i)20-s + (−2.42 + 6.55i)22-s + (2.97 + 5.14i)23-s + ⋯
L(s)  = 1  + (0.168 + 0.985i)2-s + (−0.942 + 0.333i)4-s + (−0.356 − 0.616i)5-s + (−0.471 − 0.272i)7-s + (−0.487 − 0.873i)8-s + (0.547 − 0.455i)10-s + (1.28 + 0.744i)11-s + (1.29 − 0.744i)13-s + (0.188 − 0.510i)14-s + (0.778 − 0.628i)16-s + 0.320i·17-s + 0.0614·19-s + (0.541 + 0.462i)20-s + (−0.515 + 1.39i)22-s + (0.619 + 1.07i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.653 - 0.756i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.653 - 0.756i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(648\)    =    \(2^{3} \cdot 3^{4}\)
Sign: $0.653 - 0.756i$
Analytic conductor: \(5.17430\)
Root analytic conductor: \(2.27471\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{648} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 648,\ (\ :1/2),\ 0.653 - 0.756i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.31093 + 0.599945i\)
\(L(\frac12)\) \(\approx\) \(1.31093 + 0.599945i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.238 - 1.39i)T \)
3 \( 1 \)
good5 \( 1 + (0.796 + 1.37i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (1.24 + 0.719i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-4.27 - 2.46i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-4.65 + 2.68i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 - 1.32iT - 17T^{2} \)
19 \( 1 - 0.267T + 19T^{2} \)
23 \( 1 + (-2.97 - 5.14i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.35 - 7.53i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-6.81 + 3.93i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 2.49iT - 37T^{2} \)
41 \( 1 + (-8.55 + 4.93i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (1 - 1.73i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-4.56 + 7.90i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 5.51T + 53T^{2} \)
59 \( 1 + (4.27 - 2.46i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.65 - 2.68i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.598 + 1.03i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 5.51T + 71T^{2} \)
73 \( 1 + 7.92T + 73T^{2} \)
79 \( 1 + (-10.5 - 6.09i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-6.26 - 3.61i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + 15.7iT - 89T^{2} \)
97 \( 1 + (4.69 - 8.13i)T + (-48.5 - 84.0i)T^{2} \)
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

−10.54061353023769922072870626418, −9.429564053457579172129953944675, −8.890699085933483829423709141377, −7.994232056586658271450039258575, −7.07089239309794518029511423546, −6.26983587114141787745135788289, −5.31423514992471756163152254578, −4.15945940026430312677373568433, −3.51526414038337327408323183868, −1.04771659516414692287597873291, 1.13888566248738528657286751176, 2.78063460597385498932947002702, 3.63959757312124358289091779657, 4.50062246849688883514447840641, 6.06882542674487707346578067900, 6.55802558946437887284040610678, 8.086252331748827058644586190249, 9.033005682612828708143786716801, 9.498804343748166609176190717359, 10.77770662311446617365750297481

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