↑

Properties

Label 2-7-7.3-c4-0-0
Degree $2$
Conductor $7$
Sign $0.943 - 0.331i$
Analytic cond. $0.723589$
Root an. cond. $0.850640$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (1.34 + 2.32i)2-s + (−5.53 − 3.19i)3-s + (4.38 − 7.58i)4-s + (−21.5 + 12.4i)5-s − 17.1i·6-s + (32.8 + 36.3i)7-s + 66.6·8-s + (−20.0 − 34.7i)9-s + (−58.0 − 33.5i)10-s + (−54.3 + 94.1i)11-s + (−48.5 + 28.0i)12-s − 234. i·13-s + (−40.5 + 125. i)14-s + 159.·15-s + (19.5 + 33.8i)16-s + (177. + 102. i)17-s + ⋯
L(s)  = 1  + (0.336 + 0.582i)2-s + (−0.615 − 0.355i)3-s + (0.273 − 0.474i)4-s + (−0.862 + 0.498i)5-s − 0.477i·6-s + (0.670 + 0.742i)7-s + 1.04·8-s + (−0.247 − 0.429i)9-s + (−0.580 − 0.335i)10-s + (−0.449 + 0.778i)11-s + (−0.336 + 0.194i)12-s − 1.39i·13-s + (−0.207 + 0.639i)14-s + 0.707·15-s + (0.0762 + 0.132i)16-s + (0.614 + 0.355i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.943 - 0.331i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.943 - 0.331i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7\)
Sign: $0.943 - 0.331i$
Analytic conductor: \(0.723589\)
Root analytic conductor: \(0.850640\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{7} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 7,\ (\ :2),\ 0.943 - 0.331i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.912977 + 0.155516i\)
\(L(\frac12)\) \(\approx\) \(0.912977 + 0.155516i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-32.8 - 36.3i)T \)
good2 \( 1 + (-1.34 - 2.32i)T + (-8 + 13.8i)T^{2} \)
3 \( 1 + (5.53 + 3.19i)T + (40.5 + 70.1i)T^{2} \)
5 \( 1 + (21.5 - 12.4i)T + (312.5 - 541. i)T^{2} \)
11 \( 1 + (54.3 - 94.1i)T + (-7.32e3 - 1.26e4i)T^{2} \)
13 \( 1 + 234. iT - 2.85e4T^{2} \)
17 \( 1 + (-177. - 102. i)T + (4.17e4 + 7.23e4i)T^{2} \)
19 \( 1 + (99.8 - 57.6i)T + (6.51e4 - 1.12e5i)T^{2} \)
23 \( 1 + (23.6 + 40.9i)T + (-1.39e5 + 2.42e5i)T^{2} \)
29 \( 1 + 872.T + 7.07e5T^{2} \)
31 \( 1 + (-700. - 404. i)T + (4.61e5 + 7.99e5i)T^{2} \)
37 \( 1 + (311. + 539. i)T + (-9.37e5 + 1.62e6i)T^{2} \)
41 \( 1 + 1.72e3iT - 2.82e6T^{2} \)
43 \( 1 - 2.25e3T + 3.41e6T^{2} \)
47 \( 1 + (-930. + 537. i)T + (2.43e6 - 4.22e6i)T^{2} \)
53 \( 1 + (330. - 571. i)T + (-3.94e6 - 6.83e6i)T^{2} \)
59 \( 1 + (2.52e3 + 1.45e3i)T + (6.05e6 + 1.04e7i)T^{2} \)
61 \( 1 + (1.75e3 - 1.01e3i)T + (6.92e6 - 1.19e7i)T^{2} \)
67 \( 1 + (435. - 753. i)T + (-1.00e7 - 1.74e7i)T^{2} \)
71 \( 1 + 3.71e3T + 2.54e7T^{2} \)
73 \( 1 + (-3.21e3 - 1.85e3i)T + (1.41e7 + 2.45e7i)T^{2} \)
79 \( 1 + (1.20e3 + 2.08e3i)T + (-1.94e7 + 3.37e7i)T^{2} \)
83 \( 1 - 2.66e3iT - 4.74e7T^{2} \)
89 \( 1 + (-1.11e4 + 6.43e3i)T + (3.13e7 - 5.43e7i)T^{2} \)
97 \( 1 - 9.16e3iT - 8.85e7T^{2} \)
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

−22.68254395749022146089599667832, −20.49246585482221903751467294151, −18.89164805856886392839568104445, −17.53895487750949355679892635907, −15.50514326345098503640365906933, −14.79168708061966432103980656434, −12.31923759615994163169820999040, −10.80866556962478702608479678557, −7.58222498562621002005891816579, −5.62660667520476570351731370592, 4.41042581073859805023009197156, 7.889749218084687620260149019359, 10.96948430456323517638989910064, 11.82026904593738246537811358404, 13.74224001808230354833033917117, 16.20697775890283475785899555058, 16.91753668168640460051648153884, 19.28269436842040341697999125875, 20.64830004460171205856490559458, 21.65732388870511661601314935659

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