Properties

Label 2-336-1.1-c3-0-3
Degree $2$
Conductor $336$
Sign $1$
Analytic cond. $19.8246$
Root an. cond. $4.45248$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·3-s − 21.3·5-s + 7·7-s + 9·9-s − 31.3·11-s + 84.7·13-s − 64.0·15-s − 16.0·17-s − 20·19-s + 21·21-s + 80.7·23-s + 331.·25-s + 27·27-s + 102·29-s + 245.·31-s − 94.0·33-s − 149.·35-s + 215.·37-s + 254.·39-s + 150.·41-s − 441.·43-s − 192.·45-s + 206.·47-s + 49·49-s − 48.2·51-s + 426.·53-s + 669.·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.91·5-s + 0.377·7-s + 0.333·9-s − 0.859·11-s + 1.80·13-s − 1.10·15-s − 0.229·17-s − 0.241·19-s + 0.218·21-s + 0.732·23-s + 2.64·25-s + 0.192·27-s + 0.653·29-s + 1.42·31-s − 0.496·33-s − 0.722·35-s + 0.956·37-s + 1.04·39-s + 0.574·41-s − 1.56·43-s − 0.636·45-s + 0.640·47-s + 0.142·49-s − 0.132·51-s + 1.10·53-s + 1.64·55-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $1$
Analytic conductor: \(19.8246\)
Root analytic conductor: \(4.45248\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(1.670460510\)
\(L(\frac12)\) \(\approx\) \(1.670460510\)
\(L(\frac{5}{2})\) 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 \( 1 - 7T \)
good5 \( 1 + 21.3T + 125T^{2} \)
11 \( 1 + 31.3T + 1.33e3T^{2} \)
13 \( 1 - 84.7T + 2.19e3T^{2} \)
17 \( 1 + 16.0T + 4.91e3T^{2} \)
19 \( 1 + 20T + 6.85e3T^{2} \)
23 \( 1 - 80.7T + 1.21e4T^{2} \)
29 \( 1 - 102T + 2.43e4T^{2} \)
31 \( 1 - 245.T + 2.97e4T^{2} \)
37 \( 1 - 215.T + 5.06e4T^{2} \)
41 \( 1 - 150.T + 6.89e4T^{2} \)
43 \( 1 + 441.T + 7.95e4T^{2} \)
47 \( 1 - 206.T + 1.03e5T^{2} \)
53 \( 1 - 426.T + 1.48e5T^{2} \)
59 \( 1 + 363.T + 2.05e5T^{2} \)
61 \( 1 + 343.T + 2.26e5T^{2} \)
67 \( 1 + 69.2T + 3.00e5T^{2} \)
71 \( 1 - 468.T + 3.57e5T^{2} \)
73 \( 1 - 747.T + 3.89e5T^{2} \)
79 \( 1 + 1.29e3T + 4.93e5T^{2} \)
83 \( 1 - 1.29e3T + 5.71e5T^{2} \)
89 \( 1 + 563.T + 7.04e5T^{2} \)
97 \( 1 - 1.48e3T + 9.12e5T^{2} \)
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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

−11.15129773882349163303512690122, −10.47155405386242200804040652144, −8.821555681312088065260024427185, −8.266138065637031948010448443151, −7.63537113801352281771240983655, −6.48385778391003739220078657136, −4.79298852926706830794791966290, −3.90365573406732274710843591714, −2.92264927017117321351215440460, −0.877855386743275753888098319664, 0.877855386743275753888098319664, 2.92264927017117321351215440460, 3.90365573406732274710843591714, 4.79298852926706830794791966290, 6.48385778391003739220078657136, 7.63537113801352281771240983655, 8.266138065637031948010448443151, 8.821555681312088065260024427185, 10.47155405386242200804040652144, 11.15129773882349163303512690122

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