Properties

Label 2-336-21.17-c3-0-1
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: $\chi_{336} (17, \cdot )$
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

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

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