↑

Properties

Label 2-336-7.5-c6-0-15
Degree $2$
Conductor $336$
Sign $-0.347 - 0.937i$
Analytic cond. $77.2981$
Root an. cond. $8.79193$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (13.5 − 7.79i)3-s + (214. + 123. i)5-s + (−14.2 + 342. i)7-s + (121.5 − 210. i)9-s + (−850. − 1.47e3i)11-s + 2.23e3i·13-s + 3.85e3·15-s + (−6.14e3 + 3.54e3i)17-s + (1.68e3 + 971. i)19-s + (2.47e3 + 4.73e3i)21-s + (−3.90e3 + 6.76e3i)23-s + (2.28e4 + 3.95e4i)25-s − 3.78e3i·27-s + 1.49e4·29-s + (1.06e3 − 614. i)31-s + ⋯
L(s)  = 1  + (0.5 − 0.288i)3-s + (1.71 + 0.990i)5-s + (−0.0414 + 0.999i)7-s + (0.166 − 0.288i)9-s + (−0.639 − 1.10i)11-s + 1.01i·13-s + 1.14·15-s + (−1.25 + 0.722i)17-s + (0.245 + 0.141i)19-s + (0.267 + 0.511i)21-s + (−0.321 + 0.556i)23-s + (1.46 + 2.52i)25-s − 0.192i·27-s + 0.614·29-s + (0.0357 − 0.0206i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.347 - 0.937i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.347 - 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $-0.347 - 0.937i$
Analytic conductor: \(77.2981\)
Root analytic conductor: \(8.79193\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (145, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :3),\ -0.347 - 0.937i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(2.833497026\)
\(L(\frac12)\) \(\approx\) \(2.833497026\)
\(L(4)\) 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 \( 1 + (-13.5 + 7.79i)T \)
7 \( 1 + (14.2 - 342. i)T \)
good5 \( 1 + (-214. - 123. i)T + (7.81e3 + 1.35e4i)T^{2} \)
11 \( 1 + (850. + 1.47e3i)T + (-8.85e5 + 1.53e6i)T^{2} \)
13 \( 1 - 2.23e3iT - 4.82e6T^{2} \)
17 \( 1 + (6.14e3 - 3.54e3i)T + (1.20e7 - 2.09e7i)T^{2} \)
19 \( 1 + (-1.68e3 - 971. i)T + (2.35e7 + 4.07e7i)T^{2} \)
23 \( 1 + (3.90e3 - 6.76e3i)T + (-7.40e7 - 1.28e8i)T^{2} \)
29 \( 1 - 1.49e4T + 5.94e8T^{2} \)
31 \( 1 + (-1.06e3 + 614. i)T + (4.43e8 - 7.68e8i)T^{2} \)
37 \( 1 + (999. - 1.73e3i)T + (-1.28e9 - 2.22e9i)T^{2} \)
41 \( 1 - 7.16e4iT - 4.75e9T^{2} \)
43 \( 1 + 2.44e4T + 6.32e9T^{2} \)
47 \( 1 + (6.97e4 + 4.02e4i)T + (5.38e9 + 9.33e9i)T^{2} \)
53 \( 1 + (1.13e5 + 1.96e5i)T + (-1.10e10 + 1.91e10i)T^{2} \)
59 \( 1 + (1.12e5 - 6.48e4i)T + (2.10e10 - 3.65e10i)T^{2} \)
61 \( 1 + (2.12e4 + 1.22e4i)T + (2.57e10 + 4.46e10i)T^{2} \)
67 \( 1 + (8.75e4 + 1.51e5i)T + (-4.52e10 + 7.83e10i)T^{2} \)
71 \( 1 - 3.16e5T + 1.28e11T^{2} \)
73 \( 1 + (-3.37e5 + 1.95e5i)T + (7.56e10 - 1.31e11i)T^{2} \)
79 \( 1 + (3.02e5 - 5.23e5i)T + (-1.21e11 - 2.10e11i)T^{2} \)
83 \( 1 - 3.71e5iT - 3.26e11T^{2} \)
89 \( 1 + (-1.04e6 - 6.05e5i)T + (2.48e11 + 4.30e11i)T^{2} \)
97 \( 1 + 5.60e5iT - 8.32e11T^{2} \)
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

−10.76752758366997414698495154913, −9.736411315339851720534734370231, −9.074823749800704512866525010815, −8.163412593533152134953595813920, −6.59727733007556495653464101629, −6.23877099280214876790043057903, −5.17818596041280665651735449129, −3.29097426991182555766096482196, −2.37362046641755401747587378355, −1.69006846960349958253500636440, 0.53445566931600181750675819076, 1.79386368659961870409909513670, 2.74114687963180021090301434843, 4.52516632215535352218079211752, 5.04703470818404407350938818999, 6.31677532343536924696161629857, 7.44785531657779798132315247249, 8.562130619146211475683679656274, 9.465481009354469856649076524898, 10.10691997519385154627151124613

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