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Properties

Label 2-252-63.41-c3-0-0
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: $\chi_{252} (41, \cdot )$
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

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

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