Properties

Label 2-252-12.11-c7-0-30
Degree $2$
Conductor $252$
Sign $0.635 - 0.772i$
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  + (−11.2 + 1.50i)2-s + (123. − 33.7i)4-s + 353. i·5-s − 343i·7-s + (−1.33e3 + 564. i)8-s + (−532. − 3.96e3i)10-s + 1.81e3·11-s − 1.54e4·13-s + (516. + 3.84e3i)14-s + (1.41e4 − 8.33e3i)16-s − 8.27e3i·17-s − 1.37e4i·19-s + (1.19e4 + 4.36e4i)20-s + (−2.03e4 + 2.72e3i)22-s + 9.14e4·23-s + ⋯
L(s)  = 1  + (−0.991 + 0.133i)2-s + (0.964 − 0.263i)4-s + 1.26i·5-s − 0.377i·7-s + (−0.920 + 0.389i)8-s + (−0.168 − 1.25i)10-s + 0.410·11-s − 1.94·13-s + (0.0502 + 0.374i)14-s + (0.860 − 0.508i)16-s − 0.408i·17-s − 0.460i·19-s + (0.333 + 1.22i)20-s + (−0.407 + 0.0546i)22-s + 1.56·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.635 - 0.772i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.635 - 0.772i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.635 - 0.772i$
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: \((2,\ 252,\ (\ :7/2),\ 0.635 - 0.772i)\)

Particular Values

\(L(4)\) \(\approx\) \(1.095896048\)
\(L(\frac12)\) \(\approx\) \(1.095896048\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (11.2 - 1.50i)T \)
3 \( 1 \)
7 \( 1 + 343iT \)
good5 \( 1 - 353. iT - 7.81e4T^{2} \)
11 \( 1 - 1.81e3T + 1.94e7T^{2} \)
13 \( 1 + 1.54e4T + 6.27e7T^{2} \)
17 \( 1 + 8.27e3iT - 4.10e8T^{2} \)
19 \( 1 + 1.37e4iT - 8.93e8T^{2} \)
23 \( 1 - 9.14e4T + 3.40e9T^{2} \)
29 \( 1 + 9.08e4iT - 1.72e10T^{2} \)
31 \( 1 - 2.57e4iT - 2.75e10T^{2} \)
37 \( 1 - 3.87e5T + 9.49e10T^{2} \)
41 \( 1 + 8.50e4iT - 1.94e11T^{2} \)
43 \( 1 - 3.96e5iT - 2.71e11T^{2} \)
47 \( 1 - 1.29e5T + 5.06e11T^{2} \)
53 \( 1 + 8.48e5iT - 1.17e12T^{2} \)
59 \( 1 + 7.00e5T + 2.48e12T^{2} \)
61 \( 1 + 1.61e6T + 3.14e12T^{2} \)
67 \( 1 + 3.67e6iT - 6.06e12T^{2} \)
71 \( 1 - 1.74e6T + 9.09e12T^{2} \)
73 \( 1 + 1.48e6T + 1.10e13T^{2} \)
79 \( 1 - 2.83e6iT - 1.92e13T^{2} \)
83 \( 1 - 6.73e6T + 2.71e13T^{2} \)
89 \( 1 - 1.02e7iT - 4.42e13T^{2} \)
97 \( 1 + 6.80e6T + 8.07e13T^{2} \)
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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

−10.83735227268872654585299194050, −9.876414632139243829074211874092, −9.263157541931941502060088960783, −7.74939258557787887152359511332, −7.13164748378572818956574232761, −6.42328275321912127697650124978, −4.89354455246380316823719243356, −3.08441348537435605421739923090, −2.31413064555842226573092494742, −0.67682394926351235964356184968, 0.55756445580464531966727448626, 1.62625619743287622377543650491, 2.85565070762843154735858877214, 4.54989741580635574158293335497, 5.61055902594808031118481611844, 6.98518745448239012745623765609, 7.913521029785471070143994142288, 8.960068242883705942294495252656, 9.415197606526730696342730048496, 10.45926949493547697346192829403

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