Properties

Label 2-252-21.17-c11-0-25
Degree $2$
Conductor $252$
Sign $-0.938 - 0.345i$
Analytic cond. $193.622$
Root an. cond. $13.9148$
Motivic weight $11$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (687. + 1.19e3i)5-s + (−3.74e4 − 2.40e4i)7-s + (−5.64e5 − 3.26e5i)11-s − 9.58e5i·13-s + (−3.39e6 + 5.87e6i)17-s + (1.15e7 − 6.65e6i)19-s + (3.61e7 − 2.08e7i)23-s + (2.34e7 − 4.06e7i)25-s − 1.22e8i·29-s + (−8.30e7 − 4.79e7i)31-s + (2.89e6 − 6.11e7i)35-s + (1.74e8 + 3.02e8i)37-s − 1.06e8·41-s + 8.40e8·43-s + (−2.37e8 − 4.11e8i)47-s + ⋯
L(s)  = 1  + (0.0984 + 0.170i)5-s + (−0.841 − 0.540i)7-s + (−1.05 − 0.610i)11-s − 0.715i·13-s + (−0.579 + 1.00i)17-s + (1.06 − 0.616i)19-s + (1.17 − 0.676i)23-s + (0.480 − 0.832i)25-s − 1.11i·29-s + (−0.521 − 0.300i)31-s + (0.00930 − 0.196i)35-s + (0.413 + 0.716i)37-s − 0.144·41-s + 0.871·43-s + (−0.151 − 0.261i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.938 - 0.345i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.938 - 0.345i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $-0.938 - 0.345i$
Analytic conductor: \(193.622\)
Root analytic conductor: \(13.9148\)
Motivic weight: \(11\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :11/2),\ -0.938 - 0.345i)\)

Particular Values

\(L(6)\) \(\approx\) \(0.5005505605\)
\(L(\frac12)\) \(\approx\) \(0.5005505605\)
\(L(\frac{13}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 + (3.74e4 + 2.40e4i)T \)
good5 \( 1 + (-687. - 1.19e3i)T + (-2.44e7 + 4.22e7i)T^{2} \)
11 \( 1 + (5.64e5 + 3.26e5i)T + (1.42e11 + 2.47e11i)T^{2} \)
13 \( 1 + 9.58e5iT - 1.79e12T^{2} \)
17 \( 1 + (3.39e6 - 5.87e6i)T + (-1.71e13 - 2.96e13i)T^{2} \)
19 \( 1 + (-1.15e7 + 6.65e6i)T + (5.82e13 - 1.00e14i)T^{2} \)
23 \( 1 + (-3.61e7 + 2.08e7i)T + (4.76e14 - 8.25e14i)T^{2} \)
29 \( 1 + 1.22e8iT - 1.22e16T^{2} \)
31 \( 1 + (8.30e7 + 4.79e7i)T + (1.27e16 + 2.20e16i)T^{2} \)
37 \( 1 + (-1.74e8 - 3.02e8i)T + (-8.89e16 + 1.54e17i)T^{2} \)
41 \( 1 + 1.06e8T + 5.50e17T^{2} \)
43 \( 1 - 8.40e8T + 9.29e17T^{2} \)
47 \( 1 + (2.37e8 + 4.11e8i)T + (-1.23e18 + 2.14e18i)T^{2} \)
53 \( 1 + (1.78e9 + 1.03e9i)T + (4.63e18 + 8.02e18i)T^{2} \)
59 \( 1 + (-8.22e8 + 1.42e9i)T + (-1.50e19 - 2.61e19i)T^{2} \)
61 \( 1 + (7.13e9 - 4.11e9i)T + (2.17e19 - 3.76e19i)T^{2} \)
67 \( 1 + (-8.67e9 + 1.50e10i)T + (-6.10e19 - 1.05e20i)T^{2} \)
71 \( 1 - 1.85e10iT - 2.31e20T^{2} \)
73 \( 1 + (1.63e10 + 9.46e9i)T + (1.56e20 + 2.71e20i)T^{2} \)
79 \( 1 + (-1.50e10 - 2.61e10i)T + (-3.73e20 + 6.47e20i)T^{2} \)
83 \( 1 + 7.61e9T + 1.28e21T^{2} \)
89 \( 1 + (2.98e10 + 5.17e10i)T + (-1.38e21 + 2.40e21i)T^{2} \)
97 \( 1 + 1.23e9iT - 7.15e21T^{2} \)
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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

−9.718135511689840293871550769568, −8.570498114447451896062521995331, −7.63624994837259740923696718115, −6.60299732531440660024490950092, −5.70418910345134449646153304614, −4.53249396853006619725470762706, −3.25155544399399531936181842579, −2.56253122896638068041778419640, −0.875002464596110105024954765875, −0.11125480172324162932784151984, 1.28755338921934542029792511195, 2.52780771107463892641429864928, 3.37879189334282766660935489685, 4.87590317964551034008649985829, 5.56174465534961849400230360437, 6.89494574628627313096462270751, 7.55551557161319792762881947201, 9.119049376733573528821367224574, 9.387375173867757631964562219733, 10.60651187337452909885790351494

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