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Properties

Label 2-336-21.5-c3-0-44
Degree $2$
Conductor $336$
Sign $-0.785 + 0.619i$
Analytic cond. $19.8246$
Root an. cond. $4.45248$
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  + (3.99 − 3.32i)3-s + (9.90 − 17.1i)5-s + (−18.4 + 1.84i)7-s + (4.88 − 26.5i)9-s + (−4.28 + 2.47i)11-s − 17.7i·13-s + (−17.4 − 101. i)15-s + (0.947 + 1.64i)17-s + (−83.9 − 48.4i)19-s + (−67.4 + 68.6i)21-s + (135. + 78.4i)23-s + (−133. − 231. i)25-s + (−68.7 − 122. i)27-s + 92.0i·29-s + (−67.3 + 38.8i)31-s + ⋯
L(s)  = 1  + (0.768 − 0.639i)3-s + (0.885 − 1.53i)5-s + (−0.995 + 0.0998i)7-s + (0.181 − 0.983i)9-s + (−0.117 + 0.0678i)11-s − 0.378i·13-s + (−0.301 − 1.74i)15-s + (0.0135 + 0.0234i)17-s + (−1.01 − 0.585i)19-s + (−0.700 + 0.713i)21-s + (1.23 + 0.711i)23-s + (−1.06 − 1.85i)25-s + (−0.490 − 0.871i)27-s + 0.589i·29-s + (−0.390 + 0.225i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.785 + 0.619i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.785 + 0.619i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $-0.785 + 0.619i$
Analytic conductor: \(19.8246\)
Root analytic conductor: \(4.45248\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :3/2),\ -0.785 + 0.619i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.161517776\)
\(L(\frac12)\) \(\approx\) \(2.161517776\)
\(L(\frac{5}{2})\) 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 \( 1 + (-3.99 + 3.32i)T \)
7 \( 1 + (18.4 - 1.84i)T \)
good5 \( 1 + (-9.90 + 17.1i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (4.28 - 2.47i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 17.7iT - 2.19e3T^{2} \)
17 \( 1 + (-0.947 - 1.64i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (83.9 + 48.4i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-135. - 78.4i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 92.0iT - 2.43e4T^{2} \)
31 \( 1 + (67.3 - 38.8i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (124. - 215. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 - 343.T + 6.89e4T^{2} \)
43 \( 1 - 24.5T + 7.95e4T^{2} \)
47 \( 1 + (-235. + 407. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (344. - 199. i)T + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (335. + 581. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (273. + 158. i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (116. + 202. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 152. iT - 3.57e5T^{2} \)
73 \( 1 + (-539. + 311. i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-151. + 261. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 - 856.T + 5.71e5T^{2} \)
89 \( 1 + (-453. + 785. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 - 70.3iT - 9.12e5T^{2} \)
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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

−10.55762318998261073752041882772, −9.254805546476429771831215636801, −9.186373602744594587618452802564, −8.114852061625654544722682371103, −6.87056399842660646161246814662, −5.90463698692341470398779985529, −4.78026811661565645059507863162, −3.25620177544018911243816112772, −1.93680215929909165910817602070, −0.66926604198434365824214555216, 2.26501757168426338839739971092, 3.03100776217090538939065185004, 4.12609656049023585990028164883, 5.80523305640958589181546867116, 6.66331978587121900646367171897, 7.58804906438727069524534475974, 9.035223533855836369956433274973, 9.639672611150687820720994057346, 10.60761725781280446624176709467, 10.88458955015466784531814600083

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