Properties

Label 2-252-12.11-c7-0-36
Degree $2$
Conductor $252$
Sign $0.851 - 0.523i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (9.44 − 6.23i)2-s + (50.3 − 117. i)4-s + 254. i·5-s + 343i·7-s + (−258. − 1.42e3i)8-s + (1.58e3 + 2.40e3i)10-s + 5.23e3·11-s + 0.835·13-s + (2.13e3 + 3.23e3i)14-s + (−1.13e4 − 1.18e4i)16-s + 8.42e3i·17-s + 2.02e4i·19-s + (2.99e4 + 1.27e4i)20-s + (4.94e4 − 3.26e4i)22-s − 1.03e5·23-s + ⋯
L(s)  = 1  + (0.834 − 0.550i)2-s + (0.393 − 0.919i)4-s + 0.910i·5-s + 0.377i·7-s + (−0.178 − 0.983i)8-s + (0.501 + 0.759i)10-s + 1.18·11-s + 0.000105·13-s + (0.208 + 0.315i)14-s + (−0.691 − 0.722i)16-s + 0.415i·17-s + 0.678i·19-s + (0.836 + 0.357i)20-s + (0.989 − 0.653i)22-s − 1.77·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.851 - 0.523i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.851 - 0.523i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.851 - 0.523i$
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: \((2,\ 252,\ (\ :7/2),\ 0.851 - 0.523i)\)

Particular Values

\(L(4)\) \(\approx\) \(3.458371495\)
\(L(\frac12)\) \(\approx\) \(3.458371495\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-9.44 + 6.23i)T \)
3 \( 1 \)
7 \( 1 - 343iT \)
good5 \( 1 - 254. iT - 7.81e4T^{2} \)
11 \( 1 - 5.23e3T + 1.94e7T^{2} \)
13 \( 1 - 0.835T + 6.27e7T^{2} \)
17 \( 1 - 8.42e3iT - 4.10e8T^{2} \)
19 \( 1 - 2.02e4iT - 8.93e8T^{2} \)
23 \( 1 + 1.03e5T + 3.40e9T^{2} \)
29 \( 1 + 6.10e4iT - 1.72e10T^{2} \)
31 \( 1 - 1.57e5iT - 2.75e10T^{2} \)
37 \( 1 - 4.21e5T + 9.49e10T^{2} \)
41 \( 1 - 7.63e5iT - 1.94e11T^{2} \)
43 \( 1 - 3.42e5iT - 2.71e11T^{2} \)
47 \( 1 - 1.10e6T + 5.06e11T^{2} \)
53 \( 1 - 1.42e6iT - 1.17e12T^{2} \)
59 \( 1 + 9.69e5T + 2.48e12T^{2} \)
61 \( 1 + 1.82e6T + 3.14e12T^{2} \)
67 \( 1 + 2.39e4iT - 6.06e12T^{2} \)
71 \( 1 - 3.81e6T + 9.09e12T^{2} \)
73 \( 1 - 4.49e6T + 1.10e13T^{2} \)
79 \( 1 + 5.24e5iT - 1.92e13T^{2} \)
83 \( 1 - 1.61e6T + 2.71e13T^{2} \)
89 \( 1 + 7.21e6iT - 4.42e13T^{2} \)
97 \( 1 - 7.57e6T + 8.07e13T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.02761173803366835044036110837, −10.15559757905431622090544099923, −9.275172476995164580863464857634, −7.79723797252255117570233625904, −6.43114196933185819157672807667, −6.02054852788205300665099066911, −4.44440330805020124183019601221, −3.51743996609953608776422339702, −2.41380139139893991774426082584, −1.25655556742347248851887853751, 0.61305908091236719818919969824, 2.12855299913821730808468106007, 3.75734653493585684925266338752, 4.47557439521249617400828204294, 5.59028389446109757124268913759, 6.60721190003026736581609761505, 7.61751551589866755903842023402, 8.655336535870621199999893864161, 9.506880244353196480134590916873, 11.01088587916697015830075343506

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