↑

Properties

Label 2-7e2-7.4-c5-0-0
Degree $2$
Conductor $49$
Sign $0.386 - 0.922i$
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  + (−3.54 − 6.13i)2-s + (−5.04 + 8.73i)3-s + (−9.08 + 15.7i)4-s + (−39.9 − 69.1i)5-s + 71.4·6-s − 97.9·8-s + (70.6 + 122. i)9-s + (−282. + 489. i)10-s + (−175. + 304. i)11-s + (−91.5 − 158. i)12-s + 291.·13-s + 804.·15-s + (637. + 1.10e3i)16-s + (−185. + 320. i)17-s + (500. − 866. i)18-s + (752. + 1.30e3i)19-s + ⋯
L(s)  = 1  + (−0.626 − 1.08i)2-s + (−0.323 + 0.560i)3-s + (−0.283 + 0.491i)4-s + (−0.713 − 1.23i)5-s + 0.809·6-s − 0.541·8-s + (0.290 + 0.503i)9-s + (−0.893 + 1.54i)10-s + (−0.438 + 0.759i)11-s + (−0.183 − 0.317i)12-s + 0.478·13-s + 0.923·15-s + (0.622 + 1.07i)16-s + (−0.155 + 0.268i)17-s + (0.364 − 0.630i)18-s + (0.478 + 0.828i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(49\)    =    \(7^{2}\)
Sign: $0.386 - 0.922i$
Analytic conductor: \(7.85880\)
Root analytic conductor: \(2.80335\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{49} (18, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 49,\ (\ :5/2),\ 0.386 - 0.922i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.236270 + 0.157162i\)
\(L(\frac12)\) \(\approx\) \(0.236270 + 0.157162i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
good2 \( 1 + (3.54 + 6.13i)T + (-16 + 27.7i)T^{2} \)
3 \( 1 + (5.04 - 8.73i)T + (-121.5 - 210. i)T^{2} \)
5 \( 1 + (39.9 + 69.1i)T + (-1.56e3 + 2.70e3i)T^{2} \)
11 \( 1 + (175. - 304. i)T + (-8.05e4 - 1.39e5i)T^{2} \)
13 \( 1 - 291.T + 3.71e5T^{2} \)
17 \( 1 + (185. - 320. i)T + (-7.09e5 - 1.22e6i)T^{2} \)
19 \( 1 + (-752. - 1.30e3i)T + (-1.23e6 + 2.14e6i)T^{2} \)
23 \( 1 + (-212. - 368. i)T + (-3.21e6 + 5.57e6i)T^{2} \)
29 \( 1 + 7.78e3T + 2.05e7T^{2} \)
31 \( 1 + (1.28e3 - 2.23e3i)T + (-1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 + (369. + 640. i)T + (-3.46e7 + 6.00e7i)T^{2} \)
41 \( 1 + 7.02e3T + 1.15e8T^{2} \)
43 \( 1 - 1.83e3T + 1.47e8T^{2} \)
47 \( 1 + (766. + 1.32e3i)T + (-1.14e8 + 1.98e8i)T^{2} \)
53 \( 1 + (-4.76e3 + 8.25e3i)T + (-2.09e8 - 3.62e8i)T^{2} \)
59 \( 1 + (1.48e4 - 2.56e4i)T + (-3.57e8 - 6.19e8i)T^{2} \)
61 \( 1 + (2.32e4 + 4.02e4i)T + (-4.22e8 + 7.31e8i)T^{2} \)
67 \( 1 + (1.33e4 - 2.31e4i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 + 1.43e4T + 1.80e9T^{2} \)
73 \( 1 + (3.50e4 - 6.07e4i)T + (-1.03e9 - 1.79e9i)T^{2} \)
79 \( 1 + (-1.35e4 - 2.34e4i)T + (-1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 - 7.97e4T + 3.93e9T^{2} \)
89 \( 1 + (-2.17e4 - 3.77e4i)T + (-2.79e9 + 4.83e9i)T^{2} \)
97 \( 1 + 1.03e5T + 8.58e9T^{2} \)
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

−15.15532032634864779442888769025, −13.10843893029397176363842574327, −12.22752521730632586393233858861, −11.18544620057673865382235714980, −10.14887217668862670819507522684, −9.090730502422158250625796254510, −7.84415933581969068207063320104, −5.34257159658571857758228760266, −3.92946291735235786431926917178, −1.59026187586789784826355157617, 0.18930200417393962738535795103, 3.28157187305847906012581971427, 5.96694996099664786338605598388, 7.00009296756061311574955722035, 7.72848606638037566807706835390, 9.196974905290069702508511141218, 10.92435420746236716458964668275, 11.86395887863250578171643955406, 13.41792680337770223111851925009, 14.88660459234298140187072993821

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