Properties

Label 2-336-1.1-c3-0-10
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 + 15.3·5-s + 7·7-s + 9·9-s + 5.35·11-s + 11.2·13-s + 46.0·15-s + 94.0·17-s − 20·19-s + 21·21-s − 102.·23-s + 110.·25-s + 27·27-s + 102·29-s − 341.·31-s + 16.0·33-s + 107.·35-s + 288.·37-s + 33.8·39-s − 252.·41-s + 145.·43-s + 138.·45-s + 573.·47-s + 49·49-s + 282.·51-s − 234.·53-s + 82.2·55-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.37·5-s + 0.377·7-s + 0.333·9-s + 0.146·11-s + 0.240·13-s + 0.793·15-s + 1.34·17-s − 0.241·19-s + 0.218·21-s − 0.931·23-s + 0.886·25-s + 0.192·27-s + 0.653·29-s − 1.97·31-s + 0.0847·33-s + 0.519·35-s + 1.28·37-s + 0.139·39-s − 0.963·41-s + 0.516·43-s + 0.457·45-s + 1.78·47-s + 0.142·49-s + 0.774·51-s − 0.607·53-s + 0.201·55-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(3.164878477\)
\(L(\frac12)\) \(\approx\) \(3.164878477\)
\(L(\frac{5}{2})\) 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 \( 1 - 15.3T + 125T^{2} \)
11 \( 1 - 5.35T + 1.33e3T^{2} \)
13 \( 1 - 11.2T + 2.19e3T^{2} \)
17 \( 1 - 94.0T + 4.91e3T^{2} \)
19 \( 1 + 20T + 6.85e3T^{2} \)
23 \( 1 + 102.T + 1.21e4T^{2} \)
29 \( 1 - 102T + 2.43e4T^{2} \)
31 \( 1 + 341.T + 2.97e4T^{2} \)
37 \( 1 - 288.T + 5.06e4T^{2} \)
41 \( 1 + 252.T + 6.89e4T^{2} \)
43 \( 1 - 145.T + 7.95e4T^{2} \)
47 \( 1 - 573.T + 1.03e5T^{2} \)
53 \( 1 + 234.T + 1.48e5T^{2} \)
59 \( 1 - 151.T + 2.05e5T^{2} \)
61 \( 1 - 243.T + 2.26e5T^{2} \)
67 \( 1 + 142.T + 3.00e5T^{2} \)
71 \( 1 - 65.0T + 3.57e5T^{2} \)
73 \( 1 - 380.T + 3.89e5T^{2} \)
79 \( 1 - 830.T + 4.93e5T^{2} \)
83 \( 1 + 469.T + 5.71e5T^{2} \)
89 \( 1 + 1.55e3T + 7.04e5T^{2} \)
97 \( 1 + 1.08e3T + 9.12e5T^{2} \)
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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

−10.92448015835532770483053655224, −10.01344357245228850861500545326, −9.384440304803318107321723601530, −8.391022012353445678384866354296, −7.38533854077284919410461732937, −6.11957919461537228347126185065, −5.33353502723789859113934129416, −3.86369874274752667097024434874, −2.45373406052347259862291895630, −1.38063252495230916891757415218, 1.38063252495230916891757415218, 2.45373406052347259862291895630, 3.86369874274752667097024434874, 5.33353502723789859113934129416, 6.11957919461537228347126185065, 7.38533854077284919410461732937, 8.391022012353445678384866354296, 9.384440304803318107321723601530, 10.01344357245228850861500545326, 10.92448015835532770483053655224

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