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Properties

Label 2-252-12.11-c3-0-9
Degree $2$
Conductor $252$
Sign $0.984 - 0.176i$
Analytic cond. $14.8684$
Root an. cond. $3.85596$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.614 − 2.76i)2-s + (−7.24 − 3.39i)4-s + 2.97i·5-s + 7i·7-s + (−13.8 + 17.9i)8-s + (8.22 + 1.83i)10-s − 14.5·11-s + 27.9·13-s + (19.3 + 4.30i)14-s + (40.9 + 49.1i)16-s + 47.0i·17-s + 148. i·19-s + (10.1 − 21.5i)20-s + (−8.97 + 40.2i)22-s + 85.1·23-s + ⋯
L(s)  = 1  + (0.217 − 0.976i)2-s + (−0.905 − 0.424i)4-s + 0.266i·5-s + 0.377i·7-s + (−0.611 + 0.791i)8-s + (0.260 + 0.0579i)10-s − 0.399·11-s + 0.596·13-s + (0.368 + 0.0821i)14-s + (0.639 + 0.768i)16-s + 0.671i·17-s + 1.79i·19-s + (0.113 − 0.241i)20-s + (−0.0869 + 0.390i)22-s + 0.771·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 - 0.176i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.984 - 0.176i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.984 - 0.176i$
Analytic conductor: \(14.8684\)
Root analytic conductor: \(3.85596\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :3/2),\ 0.984 - 0.176i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.492920582\)
\(L(\frac12)\) \(\approx\) \(1.492920582\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.614 + 2.76i)T \)
3 \( 1 \)
7 \( 1 - 7iT \)
good5 \( 1 - 2.97iT - 125T^{2} \)
11 \( 1 + 14.5T + 1.33e3T^{2} \)
13 \( 1 - 27.9T + 2.19e3T^{2} \)
17 \( 1 - 47.0iT - 4.91e3T^{2} \)
19 \( 1 - 148. iT - 6.85e3T^{2} \)
23 \( 1 - 85.1T + 1.21e4T^{2} \)
29 \( 1 + 197. iT - 2.43e4T^{2} \)
31 \( 1 - 213. iT - 2.97e4T^{2} \)
37 \( 1 + 3.00T + 5.06e4T^{2} \)
41 \( 1 + 4.26iT - 6.89e4T^{2} \)
43 \( 1 - 261. iT - 7.95e4T^{2} \)
47 \( 1 + 96.0T + 1.03e5T^{2} \)
53 \( 1 - 131. iT - 1.48e5T^{2} \)
59 \( 1 - 294.T + 2.05e5T^{2} \)
61 \( 1 - 134.T + 2.26e5T^{2} \)
67 \( 1 - 148. iT - 3.00e5T^{2} \)
71 \( 1 + 796.T + 3.57e5T^{2} \)
73 \( 1 + 163.T + 3.89e5T^{2} \)
79 \( 1 - 1.34e3iT - 4.93e5T^{2} \)
83 \( 1 + 534.T + 5.71e5T^{2} \)
89 \( 1 + 398. iT - 7.04e5T^{2} \)
97 \( 1 - 1.71e3T + 9.12e5T^{2} \)
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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

−11.62123270876771712844241039905, −10.67626502327179972982466815051, −10.02289293373181787908189173613, −8.835572152395854539755076198286, −8.009120821672107551331355369274, −6.32135371725789363293671455454, −5.32717097437367149822935624189, −3.98310544011385442517665269042, −2.85655940578319420028239906052, −1.40893702155469019239209040137, 0.61024180643684794592279255481, 3.08154778882098681478647947442, 4.53228726214874839459487868121, 5.35077384978667942681411958402, 6.70856811956686364594337168702, 7.40349104462048073723857221197, 8.650888050475049324525807689857, 9.264122701603509758566525096550, 10.58666023540828582210048509105, 11.63549632358229054556423987568

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