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Properties

Label 2-336-84.11-c3-0-2
Degree $2$
Conductor $336$
Sign $-0.993 - 0.109i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.993 - 0.109i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.993 - 0.109i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $-0.993 - 0.109i$
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: \((2,\ 336,\ (\ :3/2),\ -0.993 - 0.109i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.6468762186\)
\(L(\frac12)\) \(\approx\) \(0.6468762186\)
\(L(\frac{5}{2})\) 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

−11.55423894542810842600336696446, −10.26649439292259579830602664574, −9.711080495299517910412797398513, −9.009803577054582419058001923227, −7.73513522760383168826585748393, −7.03792099835506701836634906290, −5.56837222308975982988782526013, −4.37744526859110523756814770887, −3.25484379773016363779982048815, −2.18916825454981154446449886404, 0.18546518249852105098690757812, 2.14549954386262524049749659656, 3.11252100157191388777991588775, 4.40452960225702974238683716762, 5.92982150204347378637339414135, 7.06196258496928672511500572792, 7.64201023184922402301142109222, 8.955248008462439536781693478060, 9.571935007134886263920120152321, 10.42835990001576206130566617077

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