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Properties

Label 2-336-21.5-c3-0-27
Degree $2$
Conductor $336$
Sign $0.225 + 0.974i$
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  + (−0.919 + 5.11i)3-s + (−6.04 + 10.4i)5-s + (−15.9 + 9.39i)7-s + (−25.3 − 9.40i)9-s + (−30.2 + 17.4i)11-s − 81.1i·13-s + (−48.0 − 40.5i)15-s + (−4.14 − 7.17i)17-s + (52.7 + 30.4i)19-s + (−33.3 − 90.2i)21-s + (157. + 90.7i)23-s + (−10.6 − 18.4i)25-s + (71.3 − 120. i)27-s − 75.6i·29-s + (−77.3 + 44.6i)31-s + ⋯
L(s)  = 1  + (−0.176 + 0.984i)3-s + (−0.540 + 0.936i)5-s + (−0.861 + 0.507i)7-s + (−0.937 − 0.348i)9-s + (−0.829 + 0.479i)11-s − 1.73i·13-s + (−0.826 − 0.698i)15-s + (−0.0591 − 0.102i)17-s + (0.636 + 0.367i)19-s + (−0.346 − 0.938i)21-s + (1.42 + 0.822i)23-s + (−0.0850 − 0.147i)25-s + (0.508 − 0.860i)27-s − 0.484i·29-s + (−0.448 + 0.258i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.225 + 0.974i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.225 + 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.225 + 0.974i$
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: \((2,\ 336,\ (\ :3/2),\ 0.225 + 0.974i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.1020954019\)
\(L(\frac12)\) \(\approx\) \(0.1020954019\)
\(L(\frac{5}{2})\) 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 \( 1 + (0.919 - 5.11i)T \)
7 \( 1 + (15.9 - 9.39i)T \)
good5 \( 1 + (6.04 - 10.4i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (30.2 - 17.4i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 81.1iT - 2.19e3T^{2} \)
17 \( 1 + (4.14 + 7.17i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-52.7 - 30.4i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-157. - 90.7i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + 75.6iT - 2.43e4T^{2} \)
31 \( 1 + (77.3 - 44.6i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (58.4 - 101. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 - 147.T + 6.89e4T^{2} \)
43 \( 1 + 389.T + 7.95e4T^{2} \)
47 \( 1 + (99.8 - 172. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-578. + 333. i)T + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (8.54 + 14.8i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (661. + 381. i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (428. + 741. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 700. iT - 3.57e5T^{2} \)
73 \( 1 + (748. - 432. i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-463. + 802. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + 1.05e3T + 5.71e5T^{2} \)
89 \( 1 + (224. - 389. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + 213. iT - 9.12e5T^{2} \)
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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

−10.73427102736690736187250845067, −10.15478407580382154809413518741, −9.308291086557149000996809302419, −8.072292153265027862087371869825, −7.11334770263217085653909291522, −5.82082777608974179253139379644, −5.02634505538857554096981834348, −3.30948289225989499943016352190, −3.00436337905657698236694675369, −0.04207771801781048653383366371, 1.13481787289953929608007936264, 2.81472125892559712452450393950, 4.30209797011317691640491873967, 5.45803426636012699840386640774, 6.73650835334277057542179359448, 7.33096501145706208539859054273, 8.572413911401166567717440621919, 9.138401303739233352125194578939, 10.58178257930534204251920673687, 11.52907323605680188307200306339

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