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Properties

Label 2-336-84.11-c3-0-11
Degree $2$
Conductor $336$
Sign $0.592 - 0.805i$
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  + (−4.5 − 2.59i)3-s + (18.5 + 0.866i)7-s + (13.5 + 23.3i)9-s − 89·13-s + (25.5 − 14.7i)19-s + (−81 − 51.9i)21-s + (−62.5 + 108. i)25-s − 140. i·27-s + (298.5 + 172. i)31-s + (216.5 + 374. i)37-s + (400.5 + 231. i)39-s + 341. i·43-s + (341.5 + 32.0i)49-s − 153·57-s + (−91 − 157. i)61-s + ⋯
L(s)  = 1  + (−0.866 − 0.499i)3-s + (0.998 + 0.0467i)7-s + (0.5 + 0.866i)9-s − 1.89·13-s + (0.307 − 0.177i)19-s + (−0.841 − 0.539i)21-s + (−0.5 + 0.866i)25-s − 1.00i·27-s + (1.72 + 0.998i)31-s + (0.961 + 1.66i)37-s + (1.64 + 0.949i)39-s + 1.21i·43-s + (0.995 + 0.0934i)49-s − 0.355·57-s + (−0.191 − 0.330i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.592 - 0.805i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.592 - 0.805i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.592 - 0.805i$
Analytic conductor: \(19.8246\)
Root analytic conductor: \(4.45248\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (95, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :3/2),\ 0.592 - 0.805i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.120422476\)
\(L(\frac12)\) \(\approx\) \(1.120422476\)
\(L(\frac{5}{2})\) 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 \( 1 + (4.5 + 2.59i)T \)
7 \( 1 + (-18.5 - 0.866i)T \)
good5 \( 1 + (62.5 - 108. i)T^{2} \)
11 \( 1 + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 89T + 2.19e3T^{2} \)
17 \( 1 + (2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-25.5 + 14.7i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 2.43e4T^{2} \)
31 \( 1 + (-298.5 - 172. i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-216.5 - 374. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 - 6.89e4T^{2} \)
43 \( 1 - 341. iT - 7.95e4T^{2} \)
47 \( 1 + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (91 + 157. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-376.5 - 217. i)T + (1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 3.57e5T^{2} \)
73 \( 1 + (459.5 - 795. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-1.13e3 + 655. i)T + (2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + 5.71e5T^{2} \)
89 \( 1 + (3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 - 1.33e3T + 9.12e5T^{2} \)
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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

−11.57290917351419825359681594315, −10.42338070938962455338401476787, −9.613037674827023931934511654174, −8.125883075023166551603399090863, −7.46193085658857844435839194775, −6.46210648768383682061450706750, −5.15238116765068340849323056639, −4.65265098353397442479021623469, −2.55256586262323901845858281189, −1.18771820452785998687964030573, 0.50546075809287686522785139190, 2.31781472513930152821595757628, 4.18538419416570620359457716131, 4.93236119970514500835092566739, 5.89206656517005364971473508527, 7.16216285502396768188738903864, 8.025125209201770674198844258102, 9.412150247098709391269951618567, 10.11230040825301544492157071731, 11.00569490090836197114386661447

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