Properties

Label 2-252-21.17-c11-0-12
Degree $2$
Conductor $252$
Sign $0.822 + 0.569i$
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  + (3.70e3 + 6.42e3i)5-s + (−4.08e4 + 1.75e4i)7-s + (−5.92e5 − 3.42e5i)11-s + 1.28e6i·13-s + (2.64e6 − 4.57e6i)17-s + (−1.55e7 + 8.96e6i)19-s + (−3.79e6 + 2.19e6i)23-s + (−3.09e6 + 5.35e6i)25-s + 6.56e7i·29-s + (−1.57e8 − 9.09e7i)31-s + (−2.64e8 − 1.97e8i)35-s + (1.92e8 + 3.34e8i)37-s − 1.03e9·41-s + 9.00e8·43-s + (−1.08e9 − 1.87e9i)47-s + ⋯
L(s)  = 1  + (0.530 + 0.919i)5-s + (−0.918 + 0.395i)7-s + (−1.11 − 0.640i)11-s + 0.958i·13-s + (0.451 − 0.781i)17-s + (−1.43 + 0.830i)19-s + (−0.122 + 0.0709i)23-s + (−0.0633 + 0.109i)25-s + 0.594i·29-s + (−0.988 − 0.570i)31-s + (−0.851 − 0.634i)35-s + (0.457 + 0.791i)37-s − 1.39·41-s + 0.934·43-s + (−0.686 − 1.18i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.822 + 0.569i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.822 + 0.569i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.822 + 0.569i$
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: \((2,\ 252,\ (\ :11/2),\ 0.822 + 0.569i)\)

Particular Values

\(L(6)\) \(\approx\) \(0.8478216818\)
\(L(\frac12)\) \(\approx\) \(0.8478216818\)
\(L(\frac{13}{2})\) 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 \( 1 + (4.08e4 - 1.75e4i)T \)
good5 \( 1 + (-3.70e3 - 6.42e3i)T + (-2.44e7 + 4.22e7i)T^{2} \)
11 \( 1 + (5.92e5 + 3.42e5i)T + (1.42e11 + 2.47e11i)T^{2} \)
13 \( 1 - 1.28e6iT - 1.79e12T^{2} \)
17 \( 1 + (-2.64e6 + 4.57e6i)T + (-1.71e13 - 2.96e13i)T^{2} \)
19 \( 1 + (1.55e7 - 8.96e6i)T + (5.82e13 - 1.00e14i)T^{2} \)
23 \( 1 + (3.79e6 - 2.19e6i)T + (4.76e14 - 8.25e14i)T^{2} \)
29 \( 1 - 6.56e7iT - 1.22e16T^{2} \)
31 \( 1 + (1.57e8 + 9.09e7i)T + (1.27e16 + 2.20e16i)T^{2} \)
37 \( 1 + (-1.92e8 - 3.34e8i)T + (-8.89e16 + 1.54e17i)T^{2} \)
41 \( 1 + 1.03e9T + 5.50e17T^{2} \)
43 \( 1 - 9.00e8T + 9.29e17T^{2} \)
47 \( 1 + (1.08e9 + 1.87e9i)T + (-1.23e18 + 2.14e18i)T^{2} \)
53 \( 1 + (1.80e8 + 1.04e8i)T + (4.63e18 + 8.02e18i)T^{2} \)
59 \( 1 + (-1.71e9 + 2.96e9i)T + (-1.50e19 - 2.61e19i)T^{2} \)
61 \( 1 + (2.95e9 - 1.70e9i)T + (2.17e19 - 3.76e19i)T^{2} \)
67 \( 1 + (-4.27e9 + 7.40e9i)T + (-6.10e19 - 1.05e20i)T^{2} \)
71 \( 1 - 2.58e9iT - 2.31e20T^{2} \)
73 \( 1 + (5.59e9 + 3.22e9i)T + (1.56e20 + 2.71e20i)T^{2} \)
79 \( 1 + (-6.48e9 - 1.12e10i)T + (-3.73e20 + 6.47e20i)T^{2} \)
83 \( 1 - 1.06e10T + 1.28e21T^{2} \)
89 \( 1 + (-3.57e10 - 6.19e10i)T + (-1.38e21 + 2.40e21i)T^{2} \)
97 \( 1 - 5.47e10iT - 7.15e21T^{2} \)
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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

−10.10777042960710950498544762501, −9.188613464163569485294565766674, −8.103435919815219230449276435921, −6.85992171688669690182525459677, −6.21438980162931750637705759956, −5.24218941146415994578792073434, −3.70455778601732989179588435642, −2.76741589906847792834824596725, −1.96233495765183002775154887376, −0.22285838284593006389060755928, 0.61398260006418858609029578583, 1.88299789964080014146748014043, 2.97971759652654098549717237977, 4.26307709520526433383933519564, 5.28071930668795199236872563089, 6.14367325980670676719800195830, 7.34445087121412946211007901468, 8.338722365005603631830165376731, 9.304149520942165390796494254497, 10.20750060998980381574314834401

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