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Properties

Label 2-252-21.11-c8-0-20
Degree $2$
Conductor $252$
Sign $-0.994 - 0.106i$
Analytic cond. $102.659$
Root an. cond. $10.1320$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (36.7 − 21.2i)5-s + (−115. − 2.39e3i)7-s + (1.87e4 + 1.08e4i)11-s − 3.36e4·13-s + (−1.53e4 − 8.89e3i)17-s + (−5.74e4 − 9.94e4i)19-s + (7.22e4 − 4.17e4i)23-s + (−1.94e5 + 3.36e5i)25-s − 1.22e6i·29-s + (5.59e5 − 9.69e5i)31-s + (−5.52e4 − 8.57e4i)35-s + (5.32e5 + 9.22e5i)37-s + 5.39e6i·41-s + 5.04e5·43-s + (−1.18e5 + 6.83e4i)47-s + ⋯
L(s)  = 1  + (0.0588 − 0.0339i)5-s + (−0.0482 − 0.998i)7-s + (1.28 + 0.740i)11-s − 1.17·13-s + (−0.184 − 0.106i)17-s + (−0.440 − 0.763i)19-s + (0.258 − 0.149i)23-s + (−0.497 + 0.862i)25-s − 1.72i·29-s + (0.606 − 1.05i)31-s + (−0.0367 − 0.0571i)35-s + (0.284 + 0.491i)37-s + 1.90i·41-s + 0.147·43-s + (−0.0242 + 0.0140i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 - 0.106i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.994 - 0.106i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $-0.994 - 0.106i$
Analytic conductor: \(102.659\)
Root analytic conductor: \(10.1320\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :4),\ -0.994 - 0.106i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.3374679854\)
\(L(\frac12)\) \(\approx\) \(0.3374679854\)
\(L(5)\) 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 \( 1 + (115. + 2.39e3i)T \)
good5 \( 1 + (-36.7 + 21.2i)T + (1.95e5 - 3.38e5i)T^{2} \)
11 \( 1 + (-1.87e4 - 1.08e4i)T + (1.07e8 + 1.85e8i)T^{2} \)
13 \( 1 + 3.36e4T + 8.15e8T^{2} \)
17 \( 1 + (1.53e4 + 8.89e3i)T + (3.48e9 + 6.04e9i)T^{2} \)
19 \( 1 + (5.74e4 + 9.94e4i)T + (-8.49e9 + 1.47e10i)T^{2} \)
23 \( 1 + (-7.22e4 + 4.17e4i)T + (3.91e10 - 6.78e10i)T^{2} \)
29 \( 1 + 1.22e6iT - 5.00e11T^{2} \)
31 \( 1 + (-5.59e5 + 9.69e5i)T + (-4.26e11 - 7.38e11i)T^{2} \)
37 \( 1 + (-5.32e5 - 9.22e5i)T + (-1.75e12 + 3.04e12i)T^{2} \)
41 \( 1 - 5.39e6iT - 7.98e12T^{2} \)
43 \( 1 - 5.04e5T + 1.16e13T^{2} \)
47 \( 1 + (1.18e5 - 6.83e4i)T + (1.19e13 - 2.06e13i)T^{2} \)
53 \( 1 + (4.58e6 + 2.64e6i)T + (3.11e13 + 5.39e13i)T^{2} \)
59 \( 1 + (3.36e6 + 1.94e6i)T + (7.34e13 + 1.27e14i)T^{2} \)
61 \( 1 + (-7.93e6 - 1.37e7i)T + (-9.58e13 + 1.66e14i)T^{2} \)
67 \( 1 + (-1.00e7 + 1.73e7i)T + (-2.03e14 - 3.51e14i)T^{2} \)
71 \( 1 + 1.52e7iT - 6.45e14T^{2} \)
73 \( 1 + (1.41e7 - 2.44e7i)T + (-4.03e14 - 6.98e14i)T^{2} \)
79 \( 1 + (2.48e7 + 4.30e7i)T + (-7.58e14 + 1.31e15i)T^{2} \)
83 \( 1 - 2.84e7iT - 2.25e15T^{2} \)
89 \( 1 + (6.02e7 - 3.47e7i)T + (1.96e15 - 3.40e15i)T^{2} \)
97 \( 1 + 1.48e8T + 7.83e15T^{2} \)
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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

−9.807600211831608637727863500128, −9.535831532608474264830113263783, −8.013553575849010873904350544350, −7.11322020738509824279870466317, −6.31965624292391576860228585321, −4.70910633293386892147976024364, −4.07412309892947520151577674016, −2.56090133316196945100699449937, −1.27892563774203711791601147405, −0.07207722820831605913137318873, 1.46092353832518001617386596033, 2.62731656167126667983466596499, 3.82812455727722307375463120483, 5.14019362381236538446927906176, 6.12358791544323778174221694318, 7.06401012978098657448456323558, 8.447939433505822800061349036691, 9.065861267674235364249309691980, 10.08501868289523557869857876444, 11.18504148953110231166528515161

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