Properties

Label 2-252-12.11-c7-0-16
Degree $2$
Conductor $252$
Sign $0.0272 + 0.999i$
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  + (5.33 + 9.97i)2-s + (−71.0 + 106. i)4-s + 512. i·5-s − 343i·7-s + (−1.44e3 − 140. i)8-s + (−5.11e3 + 2.73e3i)10-s + 6.31e3·11-s − 5.44e3·13-s + (3.42e3 − 1.83e3i)14-s + (−6.29e3 − 1.51e4i)16-s + 3.59e4i·17-s + 3.96e4i·19-s + (−5.45e4 − 3.63e4i)20-s + (3.36e4 + 6.29e4i)22-s − 7.02e4·23-s + ⋯
L(s)  = 1  + (0.471 + 0.881i)2-s + (−0.554 + 0.831i)4-s + 1.83i·5-s − 0.377i·7-s + (−0.995 − 0.0967i)8-s + (−1.61 + 0.864i)10-s + 1.42·11-s − 0.687·13-s + (0.333 − 0.178i)14-s + (−0.384 − 0.923i)16-s + 1.77i·17-s + 1.32i·19-s + (−1.52 − 1.01i)20-s + (0.674 + 1.26i)22-s − 1.20·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0272 + 0.999i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.0272 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.0272 + 0.999i$
Analytic conductor: \(78.7210\)
Root analytic conductor: \(8.87248\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :7/2),\ 0.0272 + 0.999i)\)

Particular Values

\(L(4)\) \(\approx\) \(1.361673224\)
\(L(\frac12)\) \(\approx\) \(1.361673224\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-5.33 - 9.97i)T \)
3 \( 1 \)
7 \( 1 + 343iT \)
good5 \( 1 - 512. iT - 7.81e4T^{2} \)
11 \( 1 - 6.31e3T + 1.94e7T^{2} \)
13 \( 1 + 5.44e3T + 6.27e7T^{2} \)
17 \( 1 - 3.59e4iT - 4.10e8T^{2} \)
19 \( 1 - 3.96e4iT - 8.93e8T^{2} \)
23 \( 1 + 7.02e4T + 3.40e9T^{2} \)
29 \( 1 + 1.15e5iT - 1.72e10T^{2} \)
31 \( 1 + 8.96e4iT - 2.75e10T^{2} \)
37 \( 1 + 4.61e5T + 9.49e10T^{2} \)
41 \( 1 - 2.95e5iT - 1.94e11T^{2} \)
43 \( 1 - 1.81e5iT - 2.71e11T^{2} \)
47 \( 1 - 8.75e5T + 5.06e11T^{2} \)
53 \( 1 - 2.90e5iT - 1.17e12T^{2} \)
59 \( 1 - 1.88e6T + 2.48e12T^{2} \)
61 \( 1 + 2.05e5T + 3.14e12T^{2} \)
67 \( 1 + 4.03e6iT - 6.06e12T^{2} \)
71 \( 1 - 8.48e5T + 9.09e12T^{2} \)
73 \( 1 - 4.65e6T + 1.10e13T^{2} \)
79 \( 1 + 4.20e4iT - 1.92e13T^{2} \)
83 \( 1 + 4.15e6T + 2.71e13T^{2} \)
89 \( 1 + 7.95e6iT - 4.42e13T^{2} \)
97 \( 1 - 8.54e6T + 8.07e13T^{2} \)
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

−11.66305140213566621075388909576, −10.45893426787813600970133594192, −9.710813793193864161798844748825, −8.209991252301569931955802646998, −7.40167455821908452990766873598, −6.41461578525765341999245767359, −5.99954940761011976635556172997, −4.01388964303851127495575845581, −3.59078233353481558915729552150, −2.03340792800884477995001421869, 0.28161201099776204820903707379, 1.14125557641449780416528014242, 2.29853245361465311121968858933, 3.86798067207598061750115817023, 4.86502641833822253546715145875, 5.42244826960974806312528967033, 6.95968616751544342865385093351, 8.748355611545105945246442719015, 9.089138311493145623281744918338, 9.902344028262200574874432333458

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