↑

Properties

Label 2-336-12.11-c3-0-10
Degree $2$
Conductor $336$
Sign $0.385 - 0.922i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−4.13 + 3.15i)3-s − 1.18i·5-s + 7i·7-s + (7.12 − 26.0i)9-s + 64.7·11-s − 65.6·13-s + (3.73 + 4.88i)15-s − 19.9i·17-s − 67.2i·19-s + (−22.0 − 28.9i)21-s + 79.6·23-s + 123.·25-s + (52.7 + 130. i)27-s + 100. i·29-s + 278. i·31-s + ⋯
L(s)  = 1  + (−0.794 + 0.606i)3-s − 0.105i·5-s + 0.377i·7-s + (0.263 − 0.964i)9-s + 1.77·11-s − 1.39·13-s + (0.0642 + 0.0841i)15-s − 0.285i·17-s − 0.812i·19-s + (−0.229 − 0.300i)21-s + 0.722·23-s + 0.988·25-s + (0.375 + 0.926i)27-s + 0.643i·29-s + 1.61i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.385 - 0.922i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.385 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.385 - 0.922i$
Analytic conductor: \(19.8246\)
Root analytic conductor: \(4.45248\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :3/2),\ 0.385 - 0.922i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.353920270\)
\(L(\frac12)\) \(\approx\) \(1.353920270\)
\(L(\frac{5}{2})\) 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 \( 1 + (4.13 - 3.15i)T \)
7 \( 1 - 7iT \)
good5 \( 1 + 1.18iT - 125T^{2} \)
11 \( 1 - 64.7T + 1.33e3T^{2} \)
13 \( 1 + 65.6T + 2.19e3T^{2} \)
17 \( 1 + 19.9iT - 4.91e3T^{2} \)
19 \( 1 + 67.2iT - 6.85e3T^{2} \)
23 \( 1 - 79.6T + 1.21e4T^{2} \)
29 \( 1 - 100. iT - 2.43e4T^{2} \)
31 \( 1 - 278. iT - 2.97e4T^{2} \)
37 \( 1 + 45.5T + 5.06e4T^{2} \)
41 \( 1 - 12.4iT - 6.89e4T^{2} \)
43 \( 1 - 368. iT - 7.95e4T^{2} \)
47 \( 1 - 303.T + 1.03e5T^{2} \)
53 \( 1 - 639. iT - 1.48e5T^{2} \)
59 \( 1 + 537.T + 2.05e5T^{2} \)
61 \( 1 - 232.T + 2.26e5T^{2} \)
67 \( 1 - 533. iT - 3.00e5T^{2} \)
71 \( 1 - 1.01e3T + 3.57e5T^{2} \)
73 \( 1 + 348.T + 3.89e5T^{2} \)
79 \( 1 - 517. iT - 4.93e5T^{2} \)
83 \( 1 - 1.08e3T + 5.71e5T^{2} \)
89 \( 1 + 1.43e3iT - 7.04e5T^{2} \)
97 \( 1 - 1.76e3T + 9.12e5T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.34843619721212758663288398155, −10.44451776919297283625841410938, −9.277565009231947955572859702656, −8.993352603245530768794241632501, −7.14249719867140954083820986410, −6.50108971479181790930954751462, −5.17721183407597820191721318267, −4.47470755047209646677903810585, −3.05226484963829954436603377400, −1.08119841480815437270617945960, 0.67668771527139311584086867278, 2.05002092981949777216845353086, 3.89020008288595356641319552168, 5.02567499367967096384392996466, 6.25323276335801450369507225849, 6.97511776689817699003185261491, 7.84841808066070240775226989354, 9.194992896004497457119128761102, 10.12273285747937596224000404815, 11.10685375147153886284999796417

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