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Properties

Label 2-252-7.5-c8-0-9
Degree $2$
Conductor $252$
Sign $0.682 - 0.730i$
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  + (−632. − 365. i)5-s + (984. + 2.18e3i)7-s + (−6.35e3 − 1.10e4i)11-s + 2.11e4i·13-s + (8.25e4 − 4.76e4i)17-s + (−1.33e5 − 7.69e4i)19-s + (4.71e4 − 8.15e4i)23-s + (7.14e4 + 1.23e5i)25-s − 5.41e5·29-s + (−1.85e5 + 1.07e5i)31-s + (1.77e5 − 1.74e6i)35-s + (−3.70e5 + 6.41e5i)37-s + 7.82e5i·41-s − 2.21e5·43-s + (−7.13e6 − 4.11e6i)47-s + ⋯
L(s)  = 1  + (−1.01 − 0.584i)5-s + (0.410 + 0.912i)7-s + (−0.433 − 0.751i)11-s + 0.741i·13-s + (0.988 − 0.570i)17-s + (−1.02 − 0.590i)19-s + (0.168 − 0.291i)23-s + (0.182 + 0.316i)25-s − 0.765·29-s + (−0.201 + 0.116i)31-s + (0.117 − 1.16i)35-s + (−0.197 + 0.342i)37-s + 0.276i·41-s − 0.0647·43-s + (−1.46 − 0.843i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.682 - 0.730i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.682 - 0.730i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.682 - 0.730i$
Analytic conductor: \(102.659\)
Root analytic conductor: \(10.1320\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (145, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :4),\ 0.682 - 0.730i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(1.181174967\)
\(L(\frac12)\) \(\approx\) \(1.181174967\)
\(L(5)\) 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 \( 1 + (-984. - 2.18e3i)T \)
good5 \( 1 + (632. + 365. i)T + (1.95e5 + 3.38e5i)T^{2} \)
11 \( 1 + (6.35e3 + 1.10e4i)T + (-1.07e8 + 1.85e8i)T^{2} \)
13 \( 1 - 2.11e4iT - 8.15e8T^{2} \)
17 \( 1 + (-8.25e4 + 4.76e4i)T + (3.48e9 - 6.04e9i)T^{2} \)
19 \( 1 + (1.33e5 + 7.69e4i)T + (8.49e9 + 1.47e10i)T^{2} \)
23 \( 1 + (-4.71e4 + 8.15e4i)T + (-3.91e10 - 6.78e10i)T^{2} \)
29 \( 1 + 5.41e5T + 5.00e11T^{2} \)
31 \( 1 + (1.85e5 - 1.07e5i)T + (4.26e11 - 7.38e11i)T^{2} \)
37 \( 1 + (3.70e5 - 6.41e5i)T + (-1.75e12 - 3.04e12i)T^{2} \)
41 \( 1 - 7.82e5iT - 7.98e12T^{2} \)
43 \( 1 + 2.21e5T + 1.16e13T^{2} \)
47 \( 1 + (7.13e6 + 4.11e6i)T + (1.19e13 + 2.06e13i)T^{2} \)
53 \( 1 + (6.59e5 + 1.14e6i)T + (-3.11e13 + 5.39e13i)T^{2} \)
59 \( 1 + (-7.52e6 + 4.34e6i)T + (7.34e13 - 1.27e14i)T^{2} \)
61 \( 1 + (-1.31e7 - 7.57e6i)T + (9.58e13 + 1.66e14i)T^{2} \)
67 \( 1 + (-1.99e7 - 3.44e7i)T + (-2.03e14 + 3.51e14i)T^{2} \)
71 \( 1 - 9.34e6T + 6.45e14T^{2} \)
73 \( 1 + (-2.94e7 + 1.70e7i)T + (4.03e14 - 6.98e14i)T^{2} \)
79 \( 1 + (-2.26e7 + 3.92e7i)T + (-7.58e14 - 1.31e15i)T^{2} \)
83 \( 1 - 6.58e7iT - 2.25e15T^{2} \)
89 \( 1 + (-1.09e7 - 6.34e6i)T + (1.96e15 + 3.40e15i)T^{2} \)
97 \( 1 + 6.15e6iT - 7.83e15T^{2} \)
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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

−11.04444313035944705202995998262, −9.619854365481022733873341282999, −8.559027206882963485110866982924, −8.126697214384216532315437313370, −6.84904006827340905693143693955, −5.52827001130351864992770973362, −4.66431514455783660942668313197, −3.47692640263813365359652389313, −2.18149289677340000646789145246, −0.71658942743176952702658810784, 0.37378393798165328516909272659, 1.78937475522357218527163249118, 3.35498866577938190893101611922, 4.09532181281392645758233251674, 5.33029993382654556607437740057, 6.74622755304739046422023386092, 7.73658376601096282842967898019, 8.085075521421480760287920490309, 9.777992362166689178935887432629, 10.61610055152084843325129887066

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