Properties

Label 2-252-7.2-c11-0-2
Degree $2$
Conductor $252$
Sign $-0.909 + 0.415i$
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  + (1.33e3 − 2.32e3i)5-s + (−2.80e4 + 3.45e4i)7-s + (2.39e5 + 4.14e5i)11-s − 1.17e6·13-s + (2.40e6 + 4.17e6i)17-s + (−4.55e6 + 7.88e6i)19-s + (4.21e6 − 7.30e6i)23-s + (2.08e7 + 3.60e7i)25-s − 3.65e7·29-s + (−1.20e7 − 2.09e7i)31-s + (4.25e7 + 1.11e8i)35-s + (−1.66e8 + 2.88e8i)37-s − 5.86e7·41-s + 8.57e8·43-s + (−4.75e8 + 8.24e8i)47-s + ⋯
L(s)  = 1  + (0.191 − 0.332i)5-s + (−0.630 + 0.776i)7-s + (0.448 + 0.776i)11-s − 0.874·13-s + (0.411 + 0.712i)17-s + (−0.421 + 0.730i)19-s + (0.136 − 0.236i)23-s + (0.426 + 0.738i)25-s − 0.331·29-s + (−0.0758 − 0.131i)31-s + (0.136 + 0.358i)35-s + (−0.395 + 0.684i)37-s − 0.0791·41-s + 0.889·43-s + (−0.302 + 0.524i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.909 + 0.415i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.909 + 0.415i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $-0.909 + 0.415i$
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: \((2,\ 252,\ (\ :11/2),\ -0.909 + 0.415i)\)

Particular Values

\(L(6)\) \(\approx\) \(0.4329645579\)
\(L(\frac12)\) \(\approx\) \(0.4329645579\)
\(L(\frac{13}{2})\) 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 \( 1 + (2.80e4 - 3.45e4i)T \)
good5 \( 1 + (-1.33e3 + 2.32e3i)T + (-2.44e7 - 4.22e7i)T^{2} \)
11 \( 1 + (-2.39e5 - 4.14e5i)T + (-1.42e11 + 2.47e11i)T^{2} \)
13 \( 1 + 1.17e6T + 1.79e12T^{2} \)
17 \( 1 + (-2.40e6 - 4.17e6i)T + (-1.71e13 + 2.96e13i)T^{2} \)
19 \( 1 + (4.55e6 - 7.88e6i)T + (-5.82e13 - 1.00e14i)T^{2} \)
23 \( 1 + (-4.21e6 + 7.30e6i)T + (-4.76e14 - 8.25e14i)T^{2} \)
29 \( 1 + 3.65e7T + 1.22e16T^{2} \)
31 \( 1 + (1.20e7 + 2.09e7i)T + (-1.27e16 + 2.20e16i)T^{2} \)
37 \( 1 + (1.66e8 - 2.88e8i)T + (-8.89e16 - 1.54e17i)T^{2} \)
41 \( 1 + 5.86e7T + 5.50e17T^{2} \)
43 \( 1 - 8.57e8T + 9.29e17T^{2} \)
47 \( 1 + (4.75e8 - 8.24e8i)T + (-1.23e18 - 2.14e18i)T^{2} \)
53 \( 1 + (-9.09e8 - 1.57e9i)T + (-4.63e18 + 8.02e18i)T^{2} \)
59 \( 1 + (8.43e8 + 1.46e9i)T + (-1.50e19 + 2.61e19i)T^{2} \)
61 \( 1 + (-3.82e9 + 6.63e9i)T + (-2.17e19 - 3.76e19i)T^{2} \)
67 \( 1 + (2.92e9 + 5.06e9i)T + (-6.10e19 + 1.05e20i)T^{2} \)
71 \( 1 + 1.03e10T + 2.31e20T^{2} \)
73 \( 1 + (-4.61e9 - 7.99e9i)T + (-1.56e20 + 2.71e20i)T^{2} \)
79 \( 1 + (-9.94e9 + 1.72e10i)T + (-3.73e20 - 6.47e20i)T^{2} \)
83 \( 1 + 4.78e10T + 1.28e21T^{2} \)
89 \( 1 + (-9.70e9 + 1.68e10i)T + (-1.38e21 - 2.40e21i)T^{2} \)
97 \( 1 - 8.32e10T + 7.15e21T^{2} \)
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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

−10.44638287618238496363799039187, −9.618617598882890280925286998788, −8.912161268904668061308123416510, −7.77819537934536556334004553001, −6.68283221712383260655622436230, −5.74374352175596945017690177989, −4.74974655114531175137983415028, −3.55244365868081267767661338126, −2.35764170945433944545472741870, −1.40471053283801573920350541802, 0.088875144554082774384085114930, 0.922647806466379570406600765826, 2.44164856963942049923350134100, 3.35816421007057691968930451356, 4.45102263979308960806077684374, 5.67368699637365890148083200547, 6.76798576690229806594992863905, 7.38506193278918106518150227901, 8.720057008033691222278997783433, 9.647580598823320995417402353088

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