Properties

Label 2-252-63.20-c3-0-21
Degree $2$
Conductor $252$
Sign $-0.992 + 0.119i$
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  + (4.81 + 1.94i)3-s + (−9.12 − 15.7i)5-s + (−9.03 + 16.1i)7-s + (19.4 + 18.7i)9-s + (−49.3 − 28.4i)11-s + (−9.36 + 5.40i)13-s + (−13.2 − 93.8i)15-s − 65.2·17-s + 36.6i·19-s + (−74.9 + 60.3i)21-s + (−70.2 + 40.5i)23-s + (−103. + 179. i)25-s + (57.3 + 128. i)27-s + (−233. − 134. i)29-s + (117. − 67.9i)31-s + ⋯
L(s)  = 1  + (0.927 + 0.373i)3-s + (−0.815 − 1.41i)5-s + (−0.487 + 0.872i)7-s + (0.720 + 0.693i)9-s + (−1.35 − 0.780i)11-s + (−0.199 + 0.115i)13-s + (−0.228 − 1.61i)15-s − 0.931·17-s + 0.441i·19-s + (−0.778 + 0.627i)21-s + (−0.636 + 0.367i)23-s + (−0.830 + 1.43i)25-s + (0.408 + 0.912i)27-s + (−1.49 − 0.863i)29-s + (0.682 − 0.393i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.992 + 0.119i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.992 + 0.119i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $-0.992 + 0.119i$
Analytic conductor: \(14.8684\)
Root analytic conductor: \(3.85596\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (209, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :3/2),\ -0.992 + 0.119i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.1916771907\)
\(L(\frac12)\) \(\approx\) \(0.1916771907\)
\(L(\frac{5}{2})\) 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 \( 1 + (-4.81 - 1.94i)T \)
7 \( 1 + (9.03 - 16.1i)T \)
good5 \( 1 + (9.12 + 15.7i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (49.3 + 28.4i)T + (665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (9.36 - 5.40i)T + (1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + 65.2T + 4.91e3T^{2} \)
19 \( 1 - 36.6iT - 6.85e3T^{2} \)
23 \( 1 + (70.2 - 40.5i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (233. + 134. i)T + (1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (-117. + 67.9i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + 125.T + 5.06e4T^{2} \)
41 \( 1 + (-117. - 203. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-22.6 + 39.2i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (-241. + 417. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 - 70.1iT - 1.48e5T^{2} \)
59 \( 1 + (176. + 306. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (512. + 295. i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (261. + 453. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 895. iT - 3.57e5T^{2} \)
73 \( 1 - 982. iT - 3.89e5T^{2} \)
79 \( 1 + (-510. + 883. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (152. - 263. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 - 1.02e3T + 7.04e5T^{2} \)
97 \( 1 + (-677. - 391. i)T + (4.56e5 + 7.90e5i)T^{2} \)
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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

−11.21448293971224856470575521760, −9.897061382516427300099577139780, −9.010849605826133420417915298853, −8.332120665874899353053852674838, −7.65790972020536507031860347204, −5.75311008555655800641146390894, −4.73308830102718753826010892121, −3.60425098704697639350661383379, −2.22232426218530390649124784461, −0.06188311330007654895607340183, 2.36786955681298213087050650697, 3.30875684422840585102007664359, 4.40761341614781814515506503857, 6.49374458572642943489624173398, 7.41011572681052406907402254934, 7.67637044581489645240564269794, 9.161656613937861379028445153667, 10.39733826965549207830098558445, 10.76260153256285059361204380192, 12.19327217677904756621961230447

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