↑

Properties

Label 2-756-63.41-c3-0-20
Degree $2$
Conductor $756$
Sign $-0.482 + 0.876i$
Analytic cond. $44.6054$
Root an. cond. $6.67873$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (9.12 − 15.7i)5-s + (−9.03 − 16.1i)7-s + (49.3 − 28.4i)11-s + (−9.36 − 5.40i)13-s + 65.2·17-s − 36.6i·19-s + (70.2 + 40.5i)23-s + (−103. − 179. i)25-s + (233. − 134. i)29-s + (117. + 67.9i)31-s + (−337. − 4.68i)35-s − 125.·37-s + (−117. + 203. i)41-s + (22.6 + 39.2i)43-s + (−241. − 417. i)47-s + ⋯
L(s)  = 1  + (0.815 − 1.41i)5-s + (−0.487 − 0.872i)7-s + (1.35 − 0.780i)11-s + (−0.199 − 0.115i)13-s + 0.931·17-s − 0.441i·19-s + (0.636 + 0.367i)23-s + (−0.830 − 1.43i)25-s + (1.49 − 0.863i)29-s + (0.682 + 0.393i)31-s + (−1.63 − 0.0226i)35-s − 0.558·37-s + (−0.447 + 0.775i)41-s + (0.0802 + 0.139i)43-s + (−0.748 − 1.29i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.482 + 0.876i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.482 + 0.876i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $-0.482 + 0.876i$
Analytic conductor: \(44.6054\)
Root analytic conductor: \(6.67873\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{756} (125, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 756,\ (\ :3/2),\ -0.482 + 0.876i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.471611635\)
\(L(\frac12)\) \(\approx\) \(2.471611635\)
\(L(\frac{5}{2})\) 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 \( 1 + (9.03 + 16.1i)T \)
good5 \( 1 + (-9.12 + 15.7i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (-49.3 + 28.4i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (9.36 + 5.40i)T + (1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 - 65.2T + 4.91e3T^{2} \)
19 \( 1 + 36.6iT - 6.85e3T^{2} \)
23 \( 1 + (-70.2 - 40.5i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-233. + 134. i)T + (1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (-117. - 67.9i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + 125.T + 5.06e4T^{2} \)
41 \( 1 + (117. - 203. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-22.6 - 39.2i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (241. + 417. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 - 70.1iT - 1.48e5T^{2} \)
59 \( 1 + (-176. + 306. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (512. - 295. i)T + (1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (261. - 453. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 895. iT - 3.57e5T^{2} \)
73 \( 1 + 982. iT - 3.89e5T^{2} \)
79 \( 1 + (-510. - 883. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (-152. - 263. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + 1.02e3T + 7.04e5T^{2} \)
97 \( 1 + (-677. + 391. i)T + (4.56e5 - 7.90e5i)T^{2} \)
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

−9.691530221091721541774714656397, −8.862343365566659064669179445431, −8.187646340623162048379873138611, −6.88707440282481014511874366370, −6.12712761894152077881718342009, −5.13246114464332641156173852546, −4.24474529712012772758691429491, −3.12092423349349205885334428305, −1.35559939020157713997279314645, −0.74311132807934847218676947281, 1.56305551832371560134555204586, 2.67025264044057417140531600004, 3.47337022227160741238099741060, 4.93045707251386814327521482900, 6.15737525175841153923984261246, 6.53858114597938437773807739390, 7.39415466549931284691358091794, 8.721741204669461197895303515579, 9.574885430687749250387101128938, 10.08731608884036419693522884010

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