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Properties

Label 2-252-9.4-c3-0-1
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: $\chi_{252} (85, \cdot )$
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

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

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