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Properties

Label 2-252-7.2-c11-0-11
Degree $2$
Conductor $252$
Sign $-0.678 - 0.734i$
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  + (1.65e3 − 2.86e3i)5-s + (4.15e4 + 1.58e4i)7-s + (4.46e5 + 7.72e5i)11-s − 1.47e6·13-s + (5.35e6 + 9.26e6i)17-s + (6.82e6 − 1.18e7i)19-s + (−1.98e7 + 3.43e7i)23-s + (1.89e7 + 3.28e7i)25-s + 4.44e7·29-s + (1.82e7 + 3.16e7i)31-s + (1.14e8 − 9.28e7i)35-s + (−3.83e8 + 6.64e8i)37-s − 9.57e8·41-s − 1.31e9·43-s + (1.35e8 − 2.35e8i)47-s + ⋯
L(s)  = 1  + (0.236 − 0.410i)5-s + (0.934 + 0.356i)7-s + (0.835 + 1.44i)11-s − 1.10·13-s + (0.914 + 1.58i)17-s + (0.632 − 1.09i)19-s + (−0.641 + 1.11i)23-s + (0.387 + 0.671i)25-s + 0.402·29-s + (0.114 + 0.198i)31-s + (0.367 − 0.298i)35-s + (−0.908 + 1.57i)37-s − 1.29·41-s − 1.36·43-s + (0.0864 − 0.149i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.678 - 0.734i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.678 - 0.734i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $-0.678 - 0.734i$
Analytic conductor: \(193.622\)
Root analytic conductor: \(13.9148\)
Motivic weight: \(11\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :11/2),\ -0.678 - 0.734i)\)

Particular Values

\(L(6)\) \(\approx\) \(1.909712471\)
\(L(\frac12)\) \(\approx\) \(1.909712471\)
\(L(\frac{13}{2})\) 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 \( 1 + (-4.15e4 - 1.58e4i)T \)
good5 \( 1 + (-1.65e3 + 2.86e3i)T + (-2.44e7 - 4.22e7i)T^{2} \)
11 \( 1 + (-4.46e5 - 7.72e5i)T + (-1.42e11 + 2.47e11i)T^{2} \)
13 \( 1 + 1.47e6T + 1.79e12T^{2} \)
17 \( 1 + (-5.35e6 - 9.26e6i)T + (-1.71e13 + 2.96e13i)T^{2} \)
19 \( 1 + (-6.82e6 + 1.18e7i)T + (-5.82e13 - 1.00e14i)T^{2} \)
23 \( 1 + (1.98e7 - 3.43e7i)T + (-4.76e14 - 8.25e14i)T^{2} \)
29 \( 1 - 4.44e7T + 1.22e16T^{2} \)
31 \( 1 + (-1.82e7 - 3.16e7i)T + (-1.27e16 + 2.20e16i)T^{2} \)
37 \( 1 + (3.83e8 - 6.64e8i)T + (-8.89e16 - 1.54e17i)T^{2} \)
41 \( 1 + 9.57e8T + 5.50e17T^{2} \)
43 \( 1 + 1.31e9T + 9.29e17T^{2} \)
47 \( 1 + (-1.35e8 + 2.35e8i)T + (-1.23e18 - 2.14e18i)T^{2} \)
53 \( 1 + (2.10e9 + 3.65e9i)T + (-4.63e18 + 8.02e18i)T^{2} \)
59 \( 1 + (-5.17e8 - 8.96e8i)T + (-1.50e19 + 2.61e19i)T^{2} \)
61 \( 1 + (-1.70e9 + 2.95e9i)T + (-2.17e19 - 3.76e19i)T^{2} \)
67 \( 1 + (9.07e9 + 1.57e10i)T + (-6.10e19 + 1.05e20i)T^{2} \)
71 \( 1 + 2.82e9T + 2.31e20T^{2} \)
73 \( 1 + (-4.14e9 - 7.17e9i)T + (-1.56e20 + 2.71e20i)T^{2} \)
79 \( 1 + (4.96e9 - 8.59e9i)T + (-3.73e20 - 6.47e20i)T^{2} \)
83 \( 1 + 8.23e9T + 1.28e21T^{2} \)
89 \( 1 + (-5.05e10 + 8.76e10i)T + (-1.38e21 - 2.40e21i)T^{2} \)
97 \( 1 - 9.08e10T + 7.15e21T^{2} \)
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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

−10.22265383552933477230597576118, −9.592414051779954875635119822013, −8.545544173599899658578184544532, −7.59813515433008843779943748937, −6.65076528218909947044114582790, −5.20923589899435046175756218530, −4.75050468568810687172289391436, −3.38878032387427849262798619231, −1.81983627575540116431801128470, −1.44457299359399379921707293300, 0.33104295020701664371678596559, 1.27235062683265525688506384950, 2.55433771828505631375026077767, 3.57757762580564010211071916444, 4.81484993244320415698491080160, 5.75224934774119032630718472467, 6.89125708494791649925286145575, 7.81360948439438194734296384379, 8.728116123398512624782977284661, 9.892231193500160398590074006785

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