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Properties

Label 2-336-28.27-c3-0-0
Degree $2$
Conductor $336$
Sign $-0.733 + 0.679i$
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  − 3·3-s + 20.9i·5-s + (−17.6 − 5.46i)7-s + 9·9-s + 23.8i·11-s + 74.6i·13-s − 62.7i·15-s − 68.6i·17-s − 26.6·19-s + (53.0 + 16.4i)21-s − 74.6i·23-s − 313.·25-s − 27·27-s + 128.·29-s + 212.·31-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.87i·5-s + (−0.955 − 0.295i)7-s + 0.333·9-s + 0.653i·11-s + 1.59i·13-s − 1.08i·15-s − 0.979i·17-s − 0.321·19-s + (0.551 + 0.170i)21-s − 0.677i·23-s − 2.50·25-s − 0.192·27-s + 0.822·29-s + 1.22·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.733 + 0.679i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.733 + 0.679i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $-0.733 + 0.679i$
Analytic conductor: \(19.8246\)
Root analytic conductor: \(4.45248\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (223, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :3/2),\ -0.733 + 0.679i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.3097686574\)
\(L(\frac12)\) \(\approx\) \(0.3097686574\)
\(L(\frac{5}{2})\) 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 \( 1 + 3T \)
7 \( 1 + (17.6 + 5.46i)T \)
good5 \( 1 - 20.9iT - 125T^{2} \)
11 \( 1 - 23.8iT - 1.33e3T^{2} \)
13 \( 1 - 74.6iT - 2.19e3T^{2} \)
17 \( 1 + 68.6iT - 4.91e3T^{2} \)
19 \( 1 + 26.6T + 6.85e3T^{2} \)
23 \( 1 + 74.6iT - 1.21e4T^{2} \)
29 \( 1 - 128.T + 2.43e4T^{2} \)
31 \( 1 - 212.T + 2.97e4T^{2} \)
37 \( 1 + 329.T + 5.06e4T^{2} \)
41 \( 1 + 182. iT - 6.89e4T^{2} \)
43 \( 1 + 260. iT - 7.95e4T^{2} \)
47 \( 1 + 401.T + 1.03e5T^{2} \)
53 \( 1 + 76.7T + 1.48e5T^{2} \)
59 \( 1 + 901.T + 2.05e5T^{2} \)
61 \( 1 - 271. iT - 2.26e5T^{2} \)
67 \( 1 + 499. iT - 3.00e5T^{2} \)
71 \( 1 + 299. iT - 3.57e5T^{2} \)
73 \( 1 + 452. iT - 3.89e5T^{2} \)
79 \( 1 - 347. iT - 4.93e5T^{2} \)
83 \( 1 - 775.T + 5.71e5T^{2} \)
89 \( 1 + 48.7iT - 7.04e5T^{2} \)
97 \( 1 - 1.05e3iT - 9.12e5T^{2} \)
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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

−11.66955766036395376612647633707, −10.63634585688051439673419426233, −10.12661029926269211817102781907, −9.200199908267379383251060053355, −7.44948623623674897598299923785, −6.64639720833638875514870511781, −6.42627824969000761327152062070, −4.62463350674462104232151641453, −3.40911008751076188453989196978, −2.23811778346898725610760873930, 0.12758573228161346390308600919, 1.23966943708225325965871612758, 3.30316278952309992712604465797, 4.67055301435595551250045506519, 5.59792042540338060314688946989, 6.28114240315378803919426590599, 8.016354655256485847487123985898, 8.578940957149999053735731818669, 9.654575118779189358435029299498, 10.42289792774492653530613226212

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