↑

Properties

Label 2-252-7.4-c7-0-19
Degree $2$
Conductor $252$
Sign $-0.765 + 0.643i$
Analytic cond. $78.7210$
Root an. cond. $8.87248$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−117. − 203. i)5-s + (881. − 215. i)7-s + (3.93e3 − 6.81e3i)11-s − 5.76e3·13-s + (3.57e3 − 6.19e3i)17-s + (−1.99e4 − 3.45e4i)19-s + (2.82e4 + 4.89e4i)23-s + (1.14e4 − 1.98e4i)25-s + 1.30e5·29-s + (−2.57e4 + 4.46e4i)31-s + (−1.47e5 − 1.54e5i)35-s + (1.39e5 + 2.42e5i)37-s + 3.45e5·41-s − 1.11e5·43-s + (−4.25e5 − 7.36e5i)47-s + ⋯
L(s)  = 1  + (−0.420 − 0.728i)5-s + (0.971 − 0.237i)7-s + (0.891 − 1.54i)11-s − 0.727·13-s + (0.176 − 0.305i)17-s + (−0.667 − 1.15i)19-s + (0.484 + 0.839i)23-s + (0.146 − 0.253i)25-s + 0.990·29-s + (−0.155 + 0.269i)31-s + (−0.581 − 0.607i)35-s + (0.453 + 0.785i)37-s + 0.783·41-s − 0.214·43-s + (−0.597 − 1.03i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.765 + 0.643i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.765 + 0.643i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $-0.765 + 0.643i$
Analytic conductor: \(78.7210\)
Root analytic conductor: \(8.87248\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :7/2),\ -0.765 + 0.643i)\)

Particular Values

\(L(4)\) \(\approx\) \(1.785100096\)
\(L(\frac12)\) \(\approx\) \(1.785100096\)
\(L(\frac{9}{2})\) 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 \( 1 + (-881. + 215. i)T \)
good5 \( 1 + (117. + 203. i)T + (-3.90e4 + 6.76e4i)T^{2} \)
11 \( 1 + (-3.93e3 + 6.81e3i)T + (-9.74e6 - 1.68e7i)T^{2} \)
13 \( 1 + 5.76e3T + 6.27e7T^{2} \)
17 \( 1 + (-3.57e3 + 6.19e3i)T + (-2.05e8 - 3.55e8i)T^{2} \)
19 \( 1 + (1.99e4 + 3.45e4i)T + (-4.46e8 + 7.74e8i)T^{2} \)
23 \( 1 + (-2.82e4 - 4.89e4i)T + (-1.70e9 + 2.94e9i)T^{2} \)
29 \( 1 - 1.30e5T + 1.72e10T^{2} \)
31 \( 1 + (2.57e4 - 4.46e4i)T + (-1.37e10 - 2.38e10i)T^{2} \)
37 \( 1 + (-1.39e5 - 2.42e5i)T + (-4.74e10 + 8.22e10i)T^{2} \)
41 \( 1 - 3.45e5T + 1.94e11T^{2} \)
43 \( 1 + 1.11e5T + 2.71e11T^{2} \)
47 \( 1 + (4.25e5 + 7.36e5i)T + (-2.53e11 + 4.38e11i)T^{2} \)
53 \( 1 + (5.99e5 - 1.03e6i)T + (-5.87e11 - 1.01e12i)T^{2} \)
59 \( 1 + (8.56e5 - 1.48e6i)T + (-1.24e12 - 2.15e12i)T^{2} \)
61 \( 1 + (1.22e6 + 2.12e6i)T + (-1.57e12 + 2.72e12i)T^{2} \)
67 \( 1 + (3.13e5 - 5.43e5i)T + (-3.03e12 - 5.24e12i)T^{2} \)
71 \( 1 + 2.97e6T + 9.09e12T^{2} \)
73 \( 1 + (-4.76e5 + 8.25e5i)T + (-5.52e12 - 9.56e12i)T^{2} \)
79 \( 1 + (2.95e6 + 5.11e6i)T + (-9.60e12 + 1.66e13i)T^{2} \)
83 \( 1 - 9.78e6T + 2.71e13T^{2} \)
89 \( 1 + (2.59e6 + 4.48e6i)T + (-2.21e13 + 3.83e13i)T^{2} \)
97 \( 1 + 4.87e6T + 8.07e13T^{2} \)
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

−10.62677637221480457702082647776, −9.171041094701022327401167755328, −8.555331794266209124272000434959, −7.62717837135790377928695965636, −6.41415772204662034445076394346, −5.09241879003915498385768704868, −4.33566872327930916760680562515, −2.97072607162877584299883283719, −1.31405453113246736353126097017, −0.43940009785767311320573301581, 1.45082065343894091580955085589, 2.49578986812029802754373928388, 4.03263161365033417293403256319, 4.85179369842682378154041212489, 6.33636512895442948375202007180, 7.29424508744202359501470309002, 8.083665525933147926741593346415, 9.306049761004152126138716727666, 10.30134753841633339537953393071, 11.15733998379349530234584004388

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