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Properties

Label 2-336-21.5-c3-0-16
Degree $2$
Conductor $336$
Sign $0.903 - 0.429i$
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 − 29.4i·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 + (76.5 + 132. i)39-s + 449·43-s + (341.5 − 32.0i)49-s + 153·57-s + (810 + 467. i)61-s + ⋯
L(s)  = 1  + (−0.866 + 0.499i)3-s + (−0.998 + 0.0467i)7-s + (0.5 − 0.866i)9-s − 0.628i·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 + (0.314 + 0.544i)39-s + 1.59·43-s + (0.995 − 0.0934i)49-s + 0.355·57-s + (1.70 + 0.981i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.903 - 0.429i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.903 - 0.429i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.903 - 0.429i$
Analytic conductor: \(19.8246\)
Root analytic conductor: \(4.45248\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :3/2),\ 0.903 - 0.429i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.052193695\)
\(L(\frac12)\) \(\approx\) \(1.052193695\)
\(L(\frac{5}{2})\) 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 \( 1 + 29.4iT - 2.19e3T^{2} \)
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 \( 1 - 449T + 7.95e4T^{2} \)
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 \( 1 + (-810 - 467. i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-503.5 - 872. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 3.57e5T^{2} \)
73 \( 1 + (-730.5 + 421. i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-251.5 + 435. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + 5.71e5T^{2} \)
89 \( 1 + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + 1.37e3iT - 9.12e5T^{2} \)
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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

−11.14130036752018570764271435163, −10.20626143360082043177996149314, −9.632558061399048382202858460953, −8.510457380055838500842834489346, −7.09175492648458762585092338285, −6.24906047358422496304166192163, −5.32835139636629463882648038406, −4.13375822720980848253973000194, −2.93573807023199543799187689063, −0.74821054782764142460497242100, 0.68932070557933312828176033287, 2.36662635876735189927286967762, 3.99152180790329189962960505305, 5.23314685260740669038206188835, 6.39842052810271904728139066701, 6.87550363485240104444509645316, 8.124328974441269308574604807868, 9.306090616853137317116967001032, 10.30652703122725242881287287461, 11.02259997269818650279176482891

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