Properties

Label 2-336-12.11-c3-0-13
Degree $2$
Conductor $336$
Sign $0.836 - 0.548i$
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  + (2.33 − 4.64i)3-s + 12.0i·5-s + 7i·7-s + (−16.0 − 21.6i)9-s + 33.9·11-s − 13.8·13-s + (55.7 + 28.0i)15-s + 28.7i·17-s + 75.7i·19-s + (32.4 + 16.3i)21-s + 104.·23-s − 19.1·25-s + (−138. + 23.9i)27-s + 242. i·29-s + 316. i·31-s + ⋯
L(s)  = 1  + (0.449 − 0.893i)3-s + 1.07i·5-s + 0.377i·7-s + (−0.595 − 0.803i)9-s + 0.931·11-s − 0.295·13-s + (0.959 + 0.482i)15-s + 0.409i·17-s + 0.914i·19-s + (0.337 + 0.169i)21-s + 0.945·23-s − 0.153·25-s + (−0.985 + 0.170i)27-s + 1.55i·29-s + 1.83i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.836 - 0.548i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.836 - 0.548i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.836 - 0.548i$
Analytic conductor: \(19.8246\)
Root analytic conductor: \(4.45248\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :3/2),\ 0.836 - 0.548i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.106250018\)
\(L(\frac12)\) \(\approx\) \(2.106250018\)
\(L(\frac{5}{2})\) 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 \( 1 + (-2.33 + 4.64i)T \)
7 \( 1 - 7iT \)
good5 \( 1 - 12.0iT - 125T^{2} \)
11 \( 1 - 33.9T + 1.33e3T^{2} \)
13 \( 1 + 13.8T + 2.19e3T^{2} \)
17 \( 1 - 28.7iT - 4.91e3T^{2} \)
19 \( 1 - 75.7iT - 6.85e3T^{2} \)
23 \( 1 - 104.T + 1.21e4T^{2} \)
29 \( 1 - 242. iT - 2.43e4T^{2} \)
31 \( 1 - 316. iT - 2.97e4T^{2} \)
37 \( 1 - 303.T + 5.06e4T^{2} \)
41 \( 1 + 382. iT - 6.89e4T^{2} \)
43 \( 1 - 12.0iT - 7.95e4T^{2} \)
47 \( 1 - 314.T + 1.03e5T^{2} \)
53 \( 1 + 418. iT - 1.48e5T^{2} \)
59 \( 1 - 477.T + 2.05e5T^{2} \)
61 \( 1 - 112.T + 2.26e5T^{2} \)
67 \( 1 - 180. iT - 3.00e5T^{2} \)
71 \( 1 - 25.4T + 3.57e5T^{2} \)
73 \( 1 + 349.T + 3.89e5T^{2} \)
79 \( 1 - 452. iT - 4.93e5T^{2} \)
83 \( 1 + 1.08e3T + 5.71e5T^{2} \)
89 \( 1 - 816. iT - 7.04e5T^{2} \)
97 \( 1 + 1.07e3T + 9.12e5T^{2} \)
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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

−11.28451986209586547422085613827, −10.36936638934620875120657135637, −9.157072706563612554898094221454, −8.398860415429820020225100249233, −7.11340225759002941768702626206, −6.72282272839437980493771234129, −5.54469110633420235584706013574, −3.68671267306636620612547263629, −2.72716704069813763789155166093, −1.39668589533701214765506571054, 0.78719848659161032539630371084, 2.62759371104111275521357371814, 4.16574398290481977850046357669, 4.69236884742019546154756821925, 5.95124432423575836071757684296, 7.42030433328871038900568729420, 8.440156763587741117126765190292, 9.337525308035199845234692440815, 9.740014530800650693205700115522, 11.12035833628708112615669235454

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