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Properties

Label 2-252-21.11-c8-0-12
Degree $2$
Conductor $252$
Sign $0.910 + 0.412i$
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  + (700. − 404. i)5-s + (−641. + 2.31e3i)7-s + (−5.97e3 − 3.45e3i)11-s − 3.31e4·13-s + (−6.16e4 − 3.55e4i)17-s + (9.11e4 + 1.57e5i)19-s + (4.39e5 − 2.53e5i)23-s + (1.31e5 − 2.27e5i)25-s − 1.28e6i·29-s + (−2.20e5 + 3.82e5i)31-s + (4.86e5 + 1.87e6i)35-s + (1.41e6 + 2.44e6i)37-s − 2.61e6i·41-s + 5.88e6·43-s + (−3.57e6 + 2.06e6i)47-s + ⋯
L(s)  = 1  + (1.12 − 0.646i)5-s + (−0.267 + 0.963i)7-s + (−0.408 − 0.235i)11-s − 1.16·13-s + (−0.737 − 0.426i)17-s + (0.699 + 1.21i)19-s + (1.56 − 0.906i)23-s + (0.336 − 0.582i)25-s − 1.80i·29-s + (−0.238 + 0.413i)31-s + (0.323 + 1.25i)35-s + (0.752 + 1.30i)37-s − 0.923i·41-s + 1.72·43-s + (−0.732 + 0.422i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.910 + 0.412i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.910 + 0.412i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.910 + 0.412i$
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: \((2,\ 252,\ (\ :4),\ 0.910 + 0.412i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(2.383348204\)
\(L(\frac12)\) \(\approx\) \(2.383348204\)
\(L(5)\) 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 \( 1 + (641. - 2.31e3i)T \)
good5 \( 1 + (-700. + 404. i)T + (1.95e5 - 3.38e5i)T^{2} \)
11 \( 1 + (5.97e3 + 3.45e3i)T + (1.07e8 + 1.85e8i)T^{2} \)
13 \( 1 + 3.31e4T + 8.15e8T^{2} \)
17 \( 1 + (6.16e4 + 3.55e4i)T + (3.48e9 + 6.04e9i)T^{2} \)
19 \( 1 + (-9.11e4 - 1.57e5i)T + (-8.49e9 + 1.47e10i)T^{2} \)
23 \( 1 + (-4.39e5 + 2.53e5i)T + (3.91e10 - 6.78e10i)T^{2} \)
29 \( 1 + 1.28e6iT - 5.00e11T^{2} \)
31 \( 1 + (2.20e5 - 3.82e5i)T + (-4.26e11 - 7.38e11i)T^{2} \)
37 \( 1 + (-1.41e6 - 2.44e6i)T + (-1.75e12 + 3.04e12i)T^{2} \)
41 \( 1 + 2.61e6iT - 7.98e12T^{2} \)
43 \( 1 - 5.88e6T + 1.16e13T^{2} \)
47 \( 1 + (3.57e6 - 2.06e6i)T + (1.19e13 - 2.06e13i)T^{2} \)
53 \( 1 + (-1.19e7 - 6.88e6i)T + (3.11e13 + 5.39e13i)T^{2} \)
59 \( 1 + (-2.24e6 - 1.29e6i)T + (7.34e13 + 1.27e14i)T^{2} \)
61 \( 1 + (6.73e6 + 1.16e7i)T + (-9.58e13 + 1.66e14i)T^{2} \)
67 \( 1 + (-1.29e7 + 2.24e7i)T + (-2.03e14 - 3.51e14i)T^{2} \)
71 \( 1 + 2.46e7iT - 6.45e14T^{2} \)
73 \( 1 + (1.86e7 - 3.23e7i)T + (-4.03e14 - 6.98e14i)T^{2} \)
79 \( 1 + (1.45e7 + 2.51e7i)T + (-7.58e14 + 1.31e15i)T^{2} \)
83 \( 1 - 4.80e6iT - 2.25e15T^{2} \)
89 \( 1 + (-1.67e7 + 9.66e6i)T + (1.96e15 - 3.40e15i)T^{2} \)
97 \( 1 - 1.62e8T + 7.83e15T^{2} \)
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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

−10.31271430624205143330143479131, −9.486175387915468742764670765436, −8.886909822006292486894326614608, −7.68278107696331653519200333157, −6.32720442821186162291107356598, −5.49464550715764807375735171245, −4.69025927226257912062040285290, −2.83173603067875834712213849631, −2.06938131552397744613686973643, −0.65295517917126032423120625189, 0.822767990564858482530165830651, 2.20814349961539856103271227797, 3.12933471084703858899930405902, 4.64310953888472073291986858178, 5.63673473580283137701818259233, 7.03660602436451361807456138999, 7.24333467484023516024616001975, 9.056498857225691905209405256378, 9.770560393769968468853625182495, 10.61811214973671542971566878524

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