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Properties

Label 2-252-21.17-c11-0-16
Degree $2$
Conductor $252$
Sign $0.836 - 0.548i$
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  + (2.16e3 + 3.74e3i)5-s + (4.15e4 − 1.57e4i)7-s + (5.06e5 + 2.92e5i)11-s + 5.85e5i·13-s + (−1.10e6 + 1.90e6i)17-s + (6.80e6 − 3.92e6i)19-s + (3.45e7 − 1.99e7i)23-s + (1.50e7 − 2.60e7i)25-s + 1.26e8i·29-s + (7.34e7 + 4.23e7i)31-s + (1.48e8 + 1.21e8i)35-s + (1.67e8 + 2.89e8i)37-s + 1.08e9·41-s − 1.79e9·43-s + (−2.12e8 − 3.68e8i)47-s + ⋯
L(s)  = 1  + (0.309 + 0.536i)5-s + (0.935 − 0.353i)7-s + (0.948 + 0.547i)11-s + 0.437i·13-s + (−0.188 + 0.326i)17-s + (0.630 − 0.363i)19-s + (1.11 − 0.645i)23-s + (0.308 − 0.534i)25-s + 1.14i·29-s + (0.460 + 0.265i)31-s + (0.479 + 0.391i)35-s + (0.396 + 0.687i)37-s + 1.46·41-s − 1.85·43-s + (−0.135 − 0.234i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.836 - 0.548i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.836 - 0.548i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.836 - 0.548i$
Analytic conductor: \(193.622\)
Root analytic conductor: \(13.9148\)
Motivic weight: \(11\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :11/2),\ 0.836 - 0.548i)\)

Particular Values

\(L(6)\) \(\approx\) \(3.478337264\)
\(L(\frac12)\) \(\approx\) \(3.478337264\)
\(L(\frac{13}{2})\) 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 \( 1 + (-4.15e4 + 1.57e4i)T \)
good5 \( 1 + (-2.16e3 - 3.74e3i)T + (-2.44e7 + 4.22e7i)T^{2} \)
11 \( 1 + (-5.06e5 - 2.92e5i)T + (1.42e11 + 2.47e11i)T^{2} \)
13 \( 1 - 5.85e5iT - 1.79e12T^{2} \)
17 \( 1 + (1.10e6 - 1.90e6i)T + (-1.71e13 - 2.96e13i)T^{2} \)
19 \( 1 + (-6.80e6 + 3.92e6i)T + (5.82e13 - 1.00e14i)T^{2} \)
23 \( 1 + (-3.45e7 + 1.99e7i)T + (4.76e14 - 8.25e14i)T^{2} \)
29 \( 1 - 1.26e8iT - 1.22e16T^{2} \)
31 \( 1 + (-7.34e7 - 4.23e7i)T + (1.27e16 + 2.20e16i)T^{2} \)
37 \( 1 + (-1.67e8 - 2.89e8i)T + (-8.89e16 + 1.54e17i)T^{2} \)
41 \( 1 - 1.08e9T + 5.50e17T^{2} \)
43 \( 1 + 1.79e9T + 9.29e17T^{2} \)
47 \( 1 + (2.12e8 + 3.68e8i)T + (-1.23e18 + 2.14e18i)T^{2} \)
53 \( 1 + (-1.06e9 - 6.13e8i)T + (4.63e18 + 8.02e18i)T^{2} \)
59 \( 1 + (3.20e8 - 5.55e8i)T + (-1.50e19 - 2.61e19i)T^{2} \)
61 \( 1 + (-4.58e9 + 2.64e9i)T + (2.17e19 - 3.76e19i)T^{2} \)
67 \( 1 + (-8.81e9 + 1.52e10i)T + (-6.10e19 - 1.05e20i)T^{2} \)
71 \( 1 + 8.44e8iT - 2.31e20T^{2} \)
73 \( 1 + (9.76e9 + 5.63e9i)T + (1.56e20 + 2.71e20i)T^{2} \)
79 \( 1 + (-2.69e9 - 4.66e9i)T + (-3.73e20 + 6.47e20i)T^{2} \)
83 \( 1 + 6.89e10T + 1.28e21T^{2} \)
89 \( 1 + (1.25e10 + 2.17e10i)T + (-1.38e21 + 2.40e21i)T^{2} \)
97 \( 1 - 5.56e10iT - 7.15e21T^{2} \)
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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

−10.27933031369968415466458063625, −9.237053694775556266408456231512, −8.322849797989411128983980565146, −7.08201381214820236525301789432, −6.54265905862367146164722234756, −5.07578373464357763792001069902, −4.27491011015049596992326697374, −2.98729297678232998496576094262, −1.79752803582676840957531189537, −0.934999716463860499773343120203, 0.791977410901207303550403477540, 1.48281638041501937077745513981, 2.77013076226193065205750693189, 4.04684877491973286845079546074, 5.15188043629408998513641591569, 5.86538909287098395672339189649, 7.18174723782985915906872381800, 8.232047998256910140713625343875, 9.024831258017463443812899607988, 9.844077496266933260392991087111

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