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Properties

Label 2-252-28.27-c3-0-12
Degree $2$
Conductor $252$
Sign $0.438 - 0.898i$
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  + (−2.16 − 1.81i)2-s + (1.38 + 7.87i)4-s + 4.47i·5-s + (14.9 + 10.8i)7-s + (11.3 − 19.5i)8-s + (8.13 − 9.69i)10-s − 7.61i·11-s − 13.3i·13-s + (−12.6 − 50.8i)14-s + (−60.1 + 21.8i)16-s + 55.3i·17-s − 73.9·19-s + (−35.2 + 6.20i)20-s + (−13.8 + 16.4i)22-s + 133. i·23-s + ⋯
L(s)  = 1  + (−0.765 − 0.642i)2-s + (0.173 + 0.984i)4-s + 0.400i·5-s + (0.808 + 0.587i)7-s + (0.500 − 0.865i)8-s + (0.257 − 0.306i)10-s − 0.208i·11-s − 0.285i·13-s + (−0.241 − 0.970i)14-s + (−0.939 + 0.341i)16-s + 0.789i·17-s − 0.892·19-s + (−0.394 + 0.0693i)20-s + (−0.134 + 0.159i)22-s + 1.20i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.438 - 0.898i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.438 - 0.898i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.438 - 0.898i$
Analytic conductor: \(14.8684\)
Root analytic conductor: \(3.85596\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (55, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :3/2),\ 0.438 - 0.898i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.045670863\)
\(L(\frac12)\) \(\approx\) \(1.045670863\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2.16 + 1.81i)T \)
3 \( 1 \)
7 \( 1 + (-14.9 - 10.8i)T \)
good5 \( 1 - 4.47iT - 125T^{2} \)
11 \( 1 + 7.61iT - 1.33e3T^{2} \)
13 \( 1 + 13.3iT - 2.19e3T^{2} \)
17 \( 1 - 55.3iT - 4.91e3T^{2} \)
19 \( 1 + 73.9T + 6.85e3T^{2} \)
23 \( 1 - 133. iT - 1.21e4T^{2} \)
29 \( 1 + 23.3T + 2.43e4T^{2} \)
31 \( 1 + 241.T + 2.97e4T^{2} \)
37 \( 1 - 178.T + 5.06e4T^{2} \)
41 \( 1 - 494. iT - 6.89e4T^{2} \)
43 \( 1 + 72.6iT - 7.95e4T^{2} \)
47 \( 1 + 59.7T + 1.03e5T^{2} \)
53 \( 1 - 569.T + 1.48e5T^{2} \)
59 \( 1 + 59.5T + 2.05e5T^{2} \)
61 \( 1 - 629. iT - 2.26e5T^{2} \)
67 \( 1 - 599. iT - 3.00e5T^{2} \)
71 \( 1 + 407. iT - 3.57e5T^{2} \)
73 \( 1 - 680. iT - 3.89e5T^{2} \)
79 \( 1 - 1.08e3iT - 4.93e5T^{2} \)
83 \( 1 + 935.T + 5.71e5T^{2} \)
89 \( 1 + 12.1iT - 7.04e5T^{2} \)
97 \( 1 + 1.43e3iT - 9.12e5T^{2} \)
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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

−11.42904455774697368322328856238, −10.95061022435443626469234268499, −9.933064395610180161491318777041, −8.831628078600034385923556437262, −8.122587704278444650169013846181, −7.07556825576658845405324715642, −5.69464250584231538454707674999, −4.13117840723587735313925145610, −2.76921095322657211403204305638, −1.49087274484392956730330267081, 0.55601246272911919918225546009, 2.03646165198261998729388952231, 4.36104089853035087847274585642, 5.28970053536216368559101669043, 6.66974637892582343376304064288, 7.51539339636229148498840226578, 8.526043370738634628830247915868, 9.256791753121412012868737352825, 10.48953648767919082880844573916, 11.07101042448332758481624468996

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