↑

Properties

Label 2-756-21.11-c2-0-17
Degree $2$
Conductor $756$
Sign $-0.667 + 0.744i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (1 − 6.92i)7-s − 13-s + (−13 − 22.5i)19-s + (−12.5 + 21.6i)25-s + (6.5 − 11.2i)31-s + (−23.5 − 40.7i)37-s − 61·43-s + (−46.9 − 13.8i)49-s + (−23.5 − 40.7i)61-s + (54.5 − 94.3i)67-s + (23 − 39.8i)73-s + (−65.5 − 113. i)79-s + (−1 + 6.92i)91-s − 169·97-s + (18.5 + 32.0i)103-s + ⋯
L(s)  = 1  + (0.142 − 0.989i)7-s − 0.0769·13-s + (−0.684 − 1.18i)19-s + (−0.5 + 0.866i)25-s + (0.209 − 0.363i)31-s + (−0.635 − 1.10i)37-s − 1.41·43-s + (−0.959 − 0.282i)49-s + (−0.385 − 0.667i)61-s + (0.813 − 1.40i)67-s + (0.315 − 0.545i)73-s + (−0.829 − 1.43i)79-s + (−0.0109 + 0.0761i)91-s − 1.74·97-s + (0.179 + 0.311i)103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.667 + 0.744i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.667 + 0.744i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $-0.667 + 0.744i$
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: \((2,\ 756,\ (\ :1),\ -0.667 + 0.744i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.027723482\)
\(L(\frac12)\) \(\approx\) \(1.027723482\)
\(L(2)\) 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 \( 1 + (-1 + 6.92i)T \)
good5 \( 1 + (12.5 - 21.6i)T^{2} \)
11 \( 1 + (60.5 + 104. i)T^{2} \)
13 \( 1 + T + 169T^{2} \)
17 \( 1 + (144.5 + 250. i)T^{2} \)
19 \( 1 + (13 + 22.5i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (264.5 - 458. i)T^{2} \)
29 \( 1 - 841T^{2} \)
31 \( 1 + (-6.5 + 11.2i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + (23.5 + 40.7i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 - 1.68e3T^{2} \)
43 \( 1 + 61T + 1.84e3T^{2} \)
47 \( 1 + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (23.5 + 40.7i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-54.5 + 94.3i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 5.04e3T^{2} \)
73 \( 1 + (-23 + 39.8i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (65.5 + 113. i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 - 6.88e3T^{2} \)
89 \( 1 + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + 169T + 9.40e3T^{2} \)
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

−9.852218101883997714662923008542, −9.032956476832882567278434776893, −8.034430059985955404672914864506, −7.20905142995761025942199221336, −6.46329547012433462424245939657, −5.22025132278372397808543749675, −4.31617560369884989574250271338, −3.30518944295131613251981219807, −1.85772796488314875252834147872, −0.34186250085416878058017519204, 1.66073187639190314480181455055, 2.79595242922258626276328431217, 4.04152519567405566634751024996, 5.16331406730430559333693250274, 6.01393129278515078147971554181, 6.86781387891752114782594712489, 8.203539810912642698680908734590, 8.511536301517553243445120366144, 9.704881941825894331196619624408, 10.30984779205768055054994423992

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