↑

Properties

Label 2-252-7.5-c8-0-11
Degree $2$
Conductor $252$
Sign $-0.309 - 0.950i$
Analytic cond. $102.659$
Root an. cond. $10.1320$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−111. − 64.4i)5-s + (2.39e3 + 195. i)7-s + (1.44e4 + 2.50e4i)11-s + 3.96e4i·13-s + (6.87e4 − 3.96e4i)17-s + (6.15e4 + 3.55e4i)19-s + (−1.82e5 + 3.15e5i)23-s + (−1.87e5 − 3.23e5i)25-s − 7.65e5·29-s + (−4.99e5 + 2.88e5i)31-s + (−2.54e5 − 1.76e5i)35-s + (1.57e6 − 2.72e6i)37-s + 2.40e6i·41-s + 4.03e6·43-s + (−2.04e6 − 1.18e6i)47-s + ⋯
L(s)  = 1  + (−0.178 − 0.103i)5-s + (0.996 + 0.0816i)7-s + (0.987 + 1.71i)11-s + 1.38i·13-s + (0.822 − 0.474i)17-s + (0.472 + 0.272i)19-s + (−0.651 + 1.12i)23-s + (−0.478 − 0.829i)25-s − 1.08·29-s + (−0.540 + 0.312i)31-s + (−0.169 − 0.117i)35-s + (0.840 − 1.45i)37-s + 0.851i·41-s + 1.18·43-s + (−0.418 − 0.241i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.309 - 0.950i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.309 - 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $-0.309 - 0.950i$
Analytic conductor: \(102.659\)
Root analytic conductor: \(10.1320\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (145, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :4),\ -0.309 - 0.950i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(2.187117455\)
\(L(\frac12)\) \(\approx\) \(2.187117455\)
\(L(5)\) 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 \( 1 + (-2.39e3 - 195. i)T \)
good5 \( 1 + (111. + 64.4i)T + (1.95e5 + 3.38e5i)T^{2} \)
11 \( 1 + (-1.44e4 - 2.50e4i)T + (-1.07e8 + 1.85e8i)T^{2} \)
13 \( 1 - 3.96e4iT - 8.15e8T^{2} \)
17 \( 1 + (-6.87e4 + 3.96e4i)T + (3.48e9 - 6.04e9i)T^{2} \)
19 \( 1 + (-6.15e4 - 3.55e4i)T + (8.49e9 + 1.47e10i)T^{2} \)
23 \( 1 + (1.82e5 - 3.15e5i)T + (-3.91e10 - 6.78e10i)T^{2} \)
29 \( 1 + 7.65e5T + 5.00e11T^{2} \)
31 \( 1 + (4.99e5 - 2.88e5i)T + (4.26e11 - 7.38e11i)T^{2} \)
37 \( 1 + (-1.57e6 + 2.72e6i)T + (-1.75e12 - 3.04e12i)T^{2} \)
41 \( 1 - 2.40e6iT - 7.98e12T^{2} \)
43 \( 1 - 4.03e6T + 1.16e13T^{2} \)
47 \( 1 + (2.04e6 + 1.18e6i)T + (1.19e13 + 2.06e13i)T^{2} \)
53 \( 1 + (-1.86e6 - 3.23e6i)T + (-3.11e13 + 5.39e13i)T^{2} \)
59 \( 1 + (1.79e7 - 1.03e7i)T + (7.34e13 - 1.27e14i)T^{2} \)
61 \( 1 + (1.72e7 + 9.95e6i)T + (9.58e13 + 1.66e14i)T^{2} \)
67 \( 1 + (-4.98e6 - 8.62e6i)T + (-2.03e14 + 3.51e14i)T^{2} \)
71 \( 1 + 2.20e7T + 6.45e14T^{2} \)
73 \( 1 + (-4.52e7 + 2.60e7i)T + (4.03e14 - 6.98e14i)T^{2} \)
79 \( 1 + (1.40e7 - 2.43e7i)T + (-7.58e14 - 1.31e15i)T^{2} \)
83 \( 1 - 3.12e7iT - 2.25e15T^{2} \)
89 \( 1 + (5.95e7 + 3.43e7i)T + (1.96e15 + 3.40e15i)T^{2} \)
97 \( 1 - 9.71e7iT - 7.83e15T^{2} \)
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

−11.09977115674052975704438944297, −9.648361008209708026201796856274, −9.247760206408456316044072417870, −7.74339831145072798964108463504, −7.22376992348220131819055418660, −5.84607512187299310094060149905, −4.60385779701159020376851168990, −3.93025530383661245748401263657, −2.05012666652134539195221384148, −1.38015657862067511831156508597, 0.48406545776196157820432653833, 1.42097141891869368431049468948, 3.04627645536752097950523886824, 3.96532766618627349872665327243, 5.40312465971806429190247704558, 6.12196323616627020852676625935, 7.66580956757016519507804961157, 8.226855081006457597921702901724, 9.260642107020540215580012160821, 10.56939294123172977554497661576

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