Properties

Label 2-7e2-49.11-c5-0-9
Degree $2$
Conductor $49$
Sign $0.752 - 0.659i$
Analytic cond. $7.85880$
Root an. cond. $2.80335$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (4.21 − 3.91i)2-s + (2.43 − 0.366i)3-s + (0.0824 − 1.10i)4-s + (−33.2 + 84.6i)5-s + (8.82 − 11.0i)6-s + (122. + 42.5i)7-s + (110. + 138. i)8-s + (−226. + 69.8i)9-s + (191. + 486. i)10-s + (279. + 86.1i)11-s + (−0.202 − 2.70i)12-s + (−203. − 893. i)13-s + (683. − 299. i)14-s + (−49.7 + 218. i)15-s + (1.04e3 + 157. i)16-s + (−390. + 266. i)17-s + ⋯
L(s)  = 1  + (0.745 − 0.691i)2-s + (0.156 − 0.0235i)3-s + (0.00257 − 0.0343i)4-s + (−0.594 + 1.51i)5-s + (0.100 − 0.125i)6-s + (0.944 + 0.328i)7-s + (0.612 + 0.767i)8-s + (−0.931 + 0.287i)9-s + (0.604 + 1.53i)10-s + (0.695 + 0.214i)11-s + (−0.000406 − 0.00542i)12-s + (−0.334 − 1.46i)13-s + (0.931 − 0.408i)14-s + (−0.0571 + 0.250i)15-s + (1.02 + 0.154i)16-s + (−0.327 + 0.223i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.752 - 0.659i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.752 - 0.659i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(49\)    =    \(7^{2}\)
Sign: $0.752 - 0.659i$
Analytic conductor: \(7.85880\)
Root analytic conductor: \(2.80335\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{49} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 49,\ (\ :5/2),\ 0.752 - 0.659i)\)

Particular Values

\(L(3)\) \(\approx\) \(2.20309 + 0.828783i\)
\(L(\frac12)\) \(\approx\) \(2.20309 + 0.828783i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-122. - 42.5i)T \)
good2 \( 1 + (-4.21 + 3.91i)T + (2.39 - 31.9i)T^{2} \)
3 \( 1 + (-2.43 + 0.366i)T + (232. - 71.6i)T^{2} \)
5 \( 1 + (33.2 - 84.6i)T + (-2.29e3 - 2.12e3i)T^{2} \)
11 \( 1 + (-279. - 86.1i)T + (1.33e5 + 9.07e4i)T^{2} \)
13 \( 1 + (203. + 893. i)T + (-3.34e5 + 1.61e5i)T^{2} \)
17 \( 1 + (390. - 266. i)T + (5.18e5 - 1.32e6i)T^{2} \)
19 \( 1 + (-35.4 - 61.4i)T + (-1.23e6 + 2.14e6i)T^{2} \)
23 \( 1 + (-3.18e3 - 2.17e3i)T + (2.35e6 + 5.99e6i)T^{2} \)
29 \( 1 + (-3.43e3 - 1.65e3i)T + (1.27e7 + 1.60e7i)T^{2} \)
31 \( 1 + (-593. + 1.02e3i)T + (-1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 + (658. + 8.78e3i)T + (-6.85e7 + 1.03e7i)T^{2} \)
41 \( 1 + (7.66e3 + 9.61e3i)T + (-2.57e7 + 1.12e8i)T^{2} \)
43 \( 1 + (2.45e3 - 3.08e3i)T + (-3.27e7 - 1.43e8i)T^{2} \)
47 \( 1 + (-1.29e4 + 1.19e4i)T + (1.71e7 - 2.28e8i)T^{2} \)
53 \( 1 + (-2.00e3 + 2.67e4i)T + (-4.13e8 - 6.23e7i)T^{2} \)
59 \( 1 + (-1.52e3 - 3.87e3i)T + (-5.24e8 + 4.86e8i)T^{2} \)
61 \( 1 + (-3.39e3 - 4.53e4i)T + (-8.35e8 + 1.25e8i)T^{2} \)
67 \( 1 + (-9.26e3 + 1.60e4i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 + (-6.19e4 + 2.98e4i)T + (1.12e9 - 1.41e9i)T^{2} \)
73 \( 1 + (1.24e4 + 1.15e4i)T + (1.54e8 + 2.06e9i)T^{2} \)
79 \( 1 + (-3.69e4 - 6.40e4i)T + (-1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 + (-1.52e4 + 6.67e4i)T + (-3.54e9 - 1.70e9i)T^{2} \)
89 \( 1 + (6.42e4 - 1.98e4i)T + (4.61e9 - 3.14e9i)T^{2} \)
97 \( 1 + 5.16e4T + 8.58e9T^{2} \)
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

−14.65958643349118831892615475638, −13.69839833288885414415961247696, −12.18942852238291735139100992226, −11.27944422893610142420305699480, −10.65186698433320066040718971888, −8.401326585616859513107364374418, −7.31950122577710684373277026506, −5.32102250745250934836839966913, −3.53995833253333852058207046080, −2.48114131785985171194588307049, 1.05695840604235599580233235454, 4.28017977011152178905030430521, 5.00160426747887851869602145640, 6.68337918409096638807803684315, 8.288250389961031872424201243061, 9.233944725440056950139766864002, 11.32139485977814513499563941890, 12.26045020630727169762271034879, 13.68808865745182612854083084790, 14.39587314101529725934555468034

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