↑

Properties

Label 2-336-12.11-c1-0-4
Degree $2$
Conductor $336$
Sign $0.926 + 0.375i$
Analytic cond. $2.68297$
Root an. cond. $1.63797$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−1.06 − 1.36i)3-s + 2.12i·5-s − i·7-s + (−0.732 + 2.90i)9-s + 5.81·11-s + 4.19·13-s + (2.90 − 2.26i)15-s − 5.81i·17-s − 2.73i·19-s + (−1.36 + 1.06i)21-s − 4.25·23-s + 0.464·25-s + (4.75 − 2.09i)27-s + 5.81i·29-s + 2.53i·31-s + ⋯
L(s)  = 1  + (−0.614 − 0.788i)3-s + 0.952i·5-s − 0.377i·7-s + (−0.244 + 0.969i)9-s + 1.75·11-s + 1.16·13-s + (0.751 − 0.585i)15-s − 1.41i·17-s − 0.626i·19-s + (−0.298 + 0.232i)21-s − 0.888·23-s + 0.0928·25-s + (0.914 − 0.403i)27-s + 1.08i·29-s + 0.455i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.926 + 0.375i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.926 + 0.375i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.926 + 0.375i$
Analytic conductor: \(2.68297\)
Root analytic conductor: \(1.63797\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :1/2),\ 0.926 + 0.375i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.18100 - 0.230225i\)
\(L(\frac12)\) \(\approx\) \(1.18100 - 0.230225i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.06 + 1.36i)T \)
7 \( 1 + iT \)
good5 \( 1 - 2.12iT - 5T^{2} \)
11 \( 1 - 5.81T + 11T^{2} \)
13 \( 1 - 4.19T + 13T^{2} \)
17 \( 1 + 5.81iT - 17T^{2} \)
19 \( 1 + 2.73iT - 19T^{2} \)
23 \( 1 + 4.25T + 23T^{2} \)
29 \( 1 - 5.81iT - 29T^{2} \)
31 \( 1 - 2.53iT - 31T^{2} \)
37 \( 1 - 11.4T + 37T^{2} \)
41 \( 1 - 1.55iT - 41T^{2} \)
43 \( 1 - 2iT - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + 1.55iT - 53T^{2} \)
59 \( 1 + 9.50T + 59T^{2} \)
61 \( 1 + 1.26T + 61T^{2} \)
67 \( 1 - 3.46iT - 67T^{2} \)
71 \( 1 + 1.55T + 71T^{2} \)
73 \( 1 + 11.4T + 73T^{2} \)
79 \( 1 - 12iT - 79T^{2} \)
83 \( 1 + 9.50T + 83T^{2} \)
89 \( 1 + 13.1iT - 89T^{2} \)
97 \( 1 + 4.92T + 97T^{2} \)
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

−11.32832147425678561157120903778, −11.01055092506098433206032663908, −9.721400952860642380900626201625, −8.630219404306673968032721156825, −7.31492553754046380384057649296, −6.71842536928856475932545271068, −5.97326695759782600121999278361, −4.41615224709421167720096430448, −3.00856012859279898987969620565, −1.25528713615084735062295993055, 1.33268306698427048677751121155, 3.81520541167975223319159680453, 4.36071979960187832236372226593, 5.95712426372799149890226005004, 6.20242227917211859915451173756, 8.163936556075048517698774675164, 8.957484433441021290419974659563, 9.643842076198922595730161550664, 10.74619031227563891315961677829, 11.69724881393168504610600594115

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