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Properties

Label 2-252-21.17-c11-0-14
Degree $2$
Conductor $252$
Sign $0.924 + 0.380i$
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  + (−2.84e3 − 4.93e3i)5-s + (3.64e4 + 2.54e4i)7-s + (1.33e5 + 7.68e4i)11-s + 8.72e5i·13-s + (2.54e6 − 4.40e6i)17-s + (1.14e7 − 6.59e6i)19-s + (−2.59e7 + 1.49e7i)23-s + (8.17e6 − 1.41e7i)25-s − 9.25e7i·29-s + (−5.18e7 − 2.99e7i)31-s + (2.16e7 − 2.52e8i)35-s + (2.61e8 + 4.53e8i)37-s − 7.05e7·41-s + 1.13e9·43-s + (4.08e8 + 7.07e8i)47-s + ⋯
L(s)  = 1  + (−0.407 − 0.706i)5-s + (0.820 + 0.572i)7-s + (0.249 + 0.143i)11-s + 0.652i·13-s + (0.434 − 0.752i)17-s + (1.05 − 0.610i)19-s + (−0.839 + 0.484i)23-s + (0.167 − 0.289i)25-s − 0.838i·29-s + (−0.325 − 0.187i)31-s + (0.0697 − 0.812i)35-s + (0.621 + 1.07i)37-s − 0.0950·41-s + 1.17·43-s + (0.259 + 0.449i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.924 + 0.380i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.924 + 0.380i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.924 + 0.380i$
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: \((2,\ 252,\ (\ :11/2),\ 0.924 + 0.380i)\)

Particular Values

\(L(6)\) \(\approx\) \(2.461222557\)
\(L(\frac12)\) \(\approx\) \(2.461222557\)
\(L(\frac{13}{2})\) 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 \( 1 + (-3.64e4 - 2.54e4i)T \)
good5 \( 1 + (2.84e3 + 4.93e3i)T + (-2.44e7 + 4.22e7i)T^{2} \)
11 \( 1 + (-1.33e5 - 7.68e4i)T + (1.42e11 + 2.47e11i)T^{2} \)
13 \( 1 - 8.72e5iT - 1.79e12T^{2} \)
17 \( 1 + (-2.54e6 + 4.40e6i)T + (-1.71e13 - 2.96e13i)T^{2} \)
19 \( 1 + (-1.14e7 + 6.59e6i)T + (5.82e13 - 1.00e14i)T^{2} \)
23 \( 1 + (2.59e7 - 1.49e7i)T + (4.76e14 - 8.25e14i)T^{2} \)
29 \( 1 + 9.25e7iT - 1.22e16T^{2} \)
31 \( 1 + (5.18e7 + 2.99e7i)T + (1.27e16 + 2.20e16i)T^{2} \)
37 \( 1 + (-2.61e8 - 4.53e8i)T + (-8.89e16 + 1.54e17i)T^{2} \)
41 \( 1 + 7.05e7T + 5.50e17T^{2} \)
43 \( 1 - 1.13e9T + 9.29e17T^{2} \)
47 \( 1 + (-4.08e8 - 7.07e8i)T + (-1.23e18 + 2.14e18i)T^{2} \)
53 \( 1 + (4.61e9 + 2.66e9i)T + (4.63e18 + 8.02e18i)T^{2} \)
59 \( 1 + (1.92e9 - 3.32e9i)T + (-1.50e19 - 2.61e19i)T^{2} \)
61 \( 1 + (2.27e9 - 1.31e9i)T + (2.17e19 - 3.76e19i)T^{2} \)
67 \( 1 + (3.09e9 - 5.36e9i)T + (-6.10e19 - 1.05e20i)T^{2} \)
71 \( 1 - 1.93e10iT - 2.31e20T^{2} \)
73 \( 1 + (-1.53e10 - 8.89e9i)T + (1.56e20 + 2.71e20i)T^{2} \)
79 \( 1 + (9.58e9 + 1.66e10i)T + (-3.73e20 + 6.47e20i)T^{2} \)
83 \( 1 - 6.37e9T + 1.28e21T^{2} \)
89 \( 1 + (-7.70e9 - 1.33e10i)T + (-1.38e21 + 2.40e21i)T^{2} \)
97 \( 1 + 8.96e10iT - 7.15e21T^{2} \)
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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

−9.817751737231591447197842499973, −9.087348596456571480379258952967, −8.128195599103264854376958973775, −7.34789851724590953256132391441, −5.97219993807138325944018389250, −4.94815981560664252946273569740, −4.22474573644904241810841046381, −2.79347345555170117172818878249, −1.60551364123689584349433887081, −0.64503165636538309405438678638, 0.73675200157430633507027248301, 1.76652601011124908565818851879, 3.18770067720455736517962331598, 3.95990522070001926020736398214, 5.20845306387738190883460509539, 6.27106393384641780412527678616, 7.54371769933699661116111499167, 7.904676703044628827425338828617, 9.222068437549190644547449181838, 10.48567428891611262788517423857

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