Properties

Label 2-252-63.20-c3-0-4
Degree $2$
Conductor $252$
Sign $-0.0828 - 0.996i$
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  + (0.316 − 5.18i)3-s + (5.49 + 9.52i)5-s + (6.85 + 17.2i)7-s + (−26.8 − 3.28i)9-s + (13.8 + 7.98i)11-s + (−77.4 + 44.7i)13-s + (51.1 − 25.4i)15-s − 106.·17-s + 8.31i·19-s + (91.3 − 30.1i)21-s + (−123. + 71.5i)23-s + (2.06 − 3.58i)25-s + (−25.4 + 137. i)27-s + (129. + 74.8i)29-s + (−37.1 + 21.4i)31-s + ⋯
L(s)  = 1  + (0.0608 − 0.998i)3-s + (0.491 + 0.851i)5-s + (0.370 + 0.928i)7-s + (−0.992 − 0.121i)9-s + (0.378 + 0.218i)11-s + (−1.65 + 0.953i)13-s + (0.879 − 0.438i)15-s − 1.52·17-s + 0.100i·19-s + (0.949 − 0.313i)21-s + (−1.12 + 0.648i)23-s + (0.0165 − 0.0286i)25-s + (−0.181 + 0.983i)27-s + (0.830 + 0.479i)29-s + (−0.215 + 0.124i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0828 - 0.996i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.0828 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $-0.0828 - 0.996i$
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: \((2,\ 252,\ (\ :3/2),\ -0.0828 - 0.996i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.200379703\)
\(L(\frac12)\) \(\approx\) \(1.200379703\)
\(L(\frac{5}{2})\) 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 \( 1 + (-0.316 + 5.18i)T \)
7 \( 1 + (-6.85 - 17.2i)T \)
good5 \( 1 + (-5.49 - 9.52i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-13.8 - 7.98i)T + (665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (77.4 - 44.7i)T + (1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + 106.T + 4.91e3T^{2} \)
19 \( 1 - 8.31iT - 6.85e3T^{2} \)
23 \( 1 + (123. - 71.5i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-129. - 74.8i)T + (1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (37.1 - 21.4i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 - 390.T + 5.06e4T^{2} \)
41 \( 1 + (-172. - 298. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-28.5 + 49.4i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (-8.13 + 14.0i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + 445. iT - 1.48e5T^{2} \)
59 \( 1 + (-193. - 334. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (420. + 242. i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (251. + 436. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 751. iT - 3.57e5T^{2} \)
73 \( 1 + 507. iT - 3.89e5T^{2} \)
79 \( 1 + (-381. + 660. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (607. - 1.05e3i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 - 425.T + 7.04e5T^{2} \)
97 \( 1 + (-494. - 285. i)T + (4.56e5 + 7.90e5i)T^{2} \)
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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

−11.85901043935499541193273496135, −11.20954751912661904537494128353, −9.797906476999007682126651628518, −8.958369376049456538366478894681, −7.77161221012871008769629471219, −6.77017573675276954276842296015, −6.10660077421722982595539357439, −4.67856895763551585930080116266, −2.60905877754804987991339295534, −1.97998049698719125618679097500, 0.43693327890307940826251706425, 2.48068101119970447724486688557, 4.24569116645730382335895876747, 4.81926329469544574682354815275, 6.03282816659275282421930349267, 7.55779546507927516233589049940, 8.610363291714487101278565239948, 9.537397991127115973867380014456, 10.27724805474224288609600035596, 11.14653206823452050615777931330

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