Properties

Label 2-252-1.1-c7-0-5
Degree $2$
Conductor $252$
Sign $1$
Analytic cond. $78.7210$
Root an. cond. $8.87248$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 166.·5-s + 343·7-s − 7.80e3·11-s + 1.06e4·13-s + 7.16e3·17-s − 1.95e4·19-s + 9.62e4·23-s − 5.03e4·25-s + 5.32e4·29-s + 2.85e5·31-s + 5.71e4·35-s − 3.10e5·37-s − 6.41e5·41-s + 2.18e5·43-s + 3.94e5·47-s + 1.17e5·49-s − 1.77e6·53-s − 1.29e6·55-s + 2.93e6·59-s + 2.16e6·61-s + 1.78e6·65-s + 4.35e6·67-s − 5.14e6·71-s + 3.57e6·73-s − 2.67e6·77-s − 4.59e6·79-s + 2.98e6·83-s + ⋯
L(s)  = 1  + 0.595·5-s + 0.377·7-s − 1.76·11-s + 1.34·13-s + 0.353·17-s − 0.653·19-s + 1.64·23-s − 0.644·25-s + 0.405·29-s + 1.72·31-s + 0.225·35-s − 1.00·37-s − 1.45·41-s + 0.419·43-s + 0.554·47-s + 0.142·49-s − 1.63·53-s − 1.05·55-s + 1.86·59-s + 1.21·61-s + 0.804·65-s + 1.76·67-s − 1.70·71-s + 1.07·73-s − 0.668·77-s − 1.04·79-s + 0.572·83-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $1$
Analytic conductor: \(78.7210\)
Root analytic conductor: \(8.87248\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :7/2),\ 1)\)

Particular Values

\(L(4)\) \(\approx\) \(2.460049721\)
\(L(\frac12)\) \(\approx\) \(2.460049721\)
\(L(\frac{9}{2})\) 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 \( 1 - 343T \)
good5 \( 1 - 166.T + 7.81e4T^{2} \)
11 \( 1 + 7.80e3T + 1.94e7T^{2} \)
13 \( 1 - 1.06e4T + 6.27e7T^{2} \)
17 \( 1 - 7.16e3T + 4.10e8T^{2} \)
19 \( 1 + 1.95e4T + 8.93e8T^{2} \)
23 \( 1 - 9.62e4T + 3.40e9T^{2} \)
29 \( 1 - 5.32e4T + 1.72e10T^{2} \)
31 \( 1 - 2.85e5T + 2.75e10T^{2} \)
37 \( 1 + 3.10e5T + 9.49e10T^{2} \)
41 \( 1 + 6.41e5T + 1.94e11T^{2} \)
43 \( 1 - 2.18e5T + 2.71e11T^{2} \)
47 \( 1 - 3.94e5T + 5.06e11T^{2} \)
53 \( 1 + 1.77e6T + 1.17e12T^{2} \)
59 \( 1 - 2.93e6T + 2.48e12T^{2} \)
61 \( 1 - 2.16e6T + 3.14e12T^{2} \)
67 \( 1 - 4.35e6T + 6.06e12T^{2} \)
71 \( 1 + 5.14e6T + 9.09e12T^{2} \)
73 \( 1 - 3.57e6T + 1.10e13T^{2} \)
79 \( 1 + 4.59e6T + 1.92e13T^{2} \)
83 \( 1 - 2.98e6T + 2.71e13T^{2} \)
89 \( 1 - 2.90e6T + 4.42e13T^{2} \)
97 \( 1 - 1.21e7T + 8.07e13T^{2} \)
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

−10.65819155830517763912234262883, −10.06571837371473898083876962866, −8.692088463505091335344011255136, −8.051543659249050474528227771095, −6.75393074847147578465765611811, −5.65493113346349105300938562989, −4.81292648738004275099533490883, −3.26522037853415610984067844734, −2.11686129491500488310944246942, −0.806203463818820225274944534999, 0.806203463818820225274944534999, 2.11686129491500488310944246942, 3.26522037853415610984067844734, 4.81292648738004275099533490883, 5.65493113346349105300938562989, 6.75393074847147578465765611811, 8.051543659249050474528227771095, 8.692088463505091335344011255136, 10.06571837371473898083876962866, 10.65819155830517763912234262883

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