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Properties

Label 2-252-9.7-c3-0-17
Degree $2$
Conductor $252$
Sign $-0.488 - 0.872i$
Analytic cond. $14.8684$
Root an. cond. $3.85596$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.934 − 5.11i)3-s + (−7.13 − 12.3i)5-s + (−3.5 + 6.06i)7-s + (−25.2 − 9.55i)9-s + (−27.6 + 47.9i)11-s + (−20.0 − 34.8i)13-s + (−69.8 + 24.9i)15-s + 33.2·17-s + 31.2·19-s + (27.7 + 23.5i)21-s + (73.6 + 127. i)23-s + (−39.3 + 68.0i)25-s + (−72.4 + 120. i)27-s + (−87.4 + 151. i)29-s + (−122. − 211. i)31-s + ⋯
L(s)  = 1  + (0.179 − 0.983i)3-s + (−0.638 − 1.10i)5-s + (−0.188 + 0.327i)7-s + (−0.935 − 0.353i)9-s + (−0.758 + 1.31i)11-s + (−0.428 − 0.742i)13-s + (−1.20 + 0.428i)15-s + 0.474·17-s + 0.376·19-s + (0.287 + 0.244i)21-s + (0.667 + 1.15i)23-s + (−0.314 + 0.544i)25-s + (−0.516 + 0.856i)27-s + (−0.559 + 0.969i)29-s + (−0.707 − 1.22i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.488 - 0.872i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.488 - 0.872i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $-0.488 - 0.872i$
Analytic conductor: \(14.8684\)
Root analytic conductor: \(3.85596\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (169, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :3/2),\ -0.488 - 0.872i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.1232466046\)
\(L(\frac12)\) \(\approx\) \(0.1232466046\)
\(L(\frac{5}{2})\) 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 \( 1 + (-0.934 + 5.11i)T \)
7 \( 1 + (3.5 - 6.06i)T \)
good5 \( 1 + (7.13 + 12.3i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (27.6 - 47.9i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (20.0 + 34.8i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 - 33.2T + 4.91e3T^{2} \)
19 \( 1 - 31.2T + 6.85e3T^{2} \)
23 \( 1 + (-73.6 - 127. i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (87.4 - 151. i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (122. + 211. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 - 48.6T + 5.06e4T^{2} \)
41 \( 1 + (52.7 + 91.3i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (48.0 - 83.2i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (70.1 - 121. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + 739.T + 1.48e5T^{2} \)
59 \( 1 + (170. + 295. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (317. - 549. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (41.9 + 72.6i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 632.T + 3.57e5T^{2} \)
73 \( 1 + 877.T + 3.89e5T^{2} \)
79 \( 1 + (-473. + 820. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (325. - 563. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 - 399.T + 7.04e5T^{2} \)
97 \( 1 + (-723. + 1.25e3i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−11.26488642959181997166351198553, −9.777988112621158633473040010063, −8.904574441257712130847581506720, −7.72053552459765821098480381662, −7.41299830479260234444899414465, −5.71471885507246931987849381851, −4.81948009024849229333784972917, −3.11775454435729989878442800212, −1.58672446453187996628446499106, −0.04663245627628387731287551001, 2.85053533308346055386648452270, 3.56947833256944790488560438550, 4.88098945027168837038272695784, 6.19251648225256693478385835944, 7.39158353411891750841449381857, 8.372382361980362049907984172054, 9.473299028405919700522216424148, 10.58511161421687255683540633295, 10.97143829138805973439292751644, 11.89735599292734359180225884807

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