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Properties

Label 2-252-21.17-c11-0-13
Degree $2$
Conductor $252$
Sign $0.253 + 0.967i$
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  + (−1.04e3 − 1.81e3i)5-s + (−4.24e4 − 1.32e4i)7-s + (5.75e4 + 3.32e4i)11-s − 7.72e5i·13-s + (−5.64e6 + 9.76e6i)17-s + (−8.21e6 + 4.74e6i)19-s + (1.70e7 − 9.85e6i)23-s + (2.22e7 − 3.84e7i)25-s + 3.03e7i·29-s + (2.13e8 + 1.23e8i)31-s + (2.04e7 + 9.09e7i)35-s + (1.45e8 + 2.51e8i)37-s − 1.26e9·41-s − 1.27e9·43-s + (1.24e9 + 2.15e9i)47-s + ⋯
L(s)  = 1  + (−0.150 − 0.259i)5-s + (−0.954 − 0.297i)7-s + (0.107 + 0.0622i)11-s − 0.576i·13-s + (−0.963 + 1.66i)17-s + (−0.761 + 0.439i)19-s + (0.552 − 0.319i)23-s + (0.454 − 0.788i)25-s + 0.275i·29-s + (1.34 + 0.773i)31-s + (0.0658 + 0.292i)35-s + (0.344 + 0.596i)37-s − 1.69·41-s − 1.31·43-s + (0.791 + 1.37i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.253 + 0.967i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.253 + 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.253 + 0.967i$
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: \((2,\ 252,\ (\ :11/2),\ 0.253 + 0.967i)\)

Particular Values

\(L(6)\) \(\approx\) \(1.014973854\)
\(L(\frac12)\) \(\approx\) \(1.014973854\)
\(L(\frac{13}{2})\) 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 \( 1 + (4.24e4 + 1.32e4i)T \)
good5 \( 1 + (1.04e3 + 1.81e3i)T + (-2.44e7 + 4.22e7i)T^{2} \)
11 \( 1 + (-5.75e4 - 3.32e4i)T + (1.42e11 + 2.47e11i)T^{2} \)
13 \( 1 + 7.72e5iT - 1.79e12T^{2} \)
17 \( 1 + (5.64e6 - 9.76e6i)T + (-1.71e13 - 2.96e13i)T^{2} \)
19 \( 1 + (8.21e6 - 4.74e6i)T + (5.82e13 - 1.00e14i)T^{2} \)
23 \( 1 + (-1.70e7 + 9.85e6i)T + (4.76e14 - 8.25e14i)T^{2} \)
29 \( 1 - 3.03e7iT - 1.22e16T^{2} \)
31 \( 1 + (-2.13e8 - 1.23e8i)T + (1.27e16 + 2.20e16i)T^{2} \)
37 \( 1 + (-1.45e8 - 2.51e8i)T + (-8.89e16 + 1.54e17i)T^{2} \)
41 \( 1 + 1.26e9T + 5.50e17T^{2} \)
43 \( 1 + 1.27e9T + 9.29e17T^{2} \)
47 \( 1 + (-1.24e9 - 2.15e9i)T + (-1.23e18 + 2.14e18i)T^{2} \)
53 \( 1 + (-4.41e8 - 2.55e8i)T + (4.63e18 + 8.02e18i)T^{2} \)
59 \( 1 + (-3.93e9 + 6.81e9i)T + (-1.50e19 - 2.61e19i)T^{2} \)
61 \( 1 + (-8.63e9 + 4.98e9i)T + (2.17e19 - 3.76e19i)T^{2} \)
67 \( 1 + (9.65e9 - 1.67e10i)T + (-6.10e19 - 1.05e20i)T^{2} \)
71 \( 1 + 3.06e9iT - 2.31e20T^{2} \)
73 \( 1 + (1.57e10 + 9.11e9i)T + (1.56e20 + 2.71e20i)T^{2} \)
79 \( 1 + (1.59e10 + 2.76e10i)T + (-3.73e20 + 6.47e20i)T^{2} \)
83 \( 1 - 1.76e10T + 1.28e21T^{2} \)
89 \( 1 + (1.13e10 + 1.96e10i)T + (-1.38e21 + 2.40e21i)T^{2} \)
97 \( 1 - 1.30e10iT - 7.15e21T^{2} \)
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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

−10.16858433654232565786091386533, −8.771204883566631485501535520532, −8.208450576662014588364602597734, −6.73334933771842533673635914982, −6.22611933255670512928744698956, −4.78848106679594269265518056041, −3.83088327457040409799393071378, −2.78915535659506219728605892979, −1.45973350974406553709236946021, −0.27664293714845416896099019634, 0.65091548728785121141512474660, 2.23917264346022872915186015172, 3.06518450436501309512254178678, 4.25053369886965654757161733386, 5.35558744531157023087648608688, 6.68299686744544663430607763204, 7.05191080537698623623305193813, 8.609223247639869180714643664574, 9.308279289934037372061368068916, 10.20798442632447353770510915952

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