↑

Properties

Label 2-168-7.4-c9-0-12
Degree $2$
Conductor $168$
Sign $0.366 - 0.930i$
Analytic cond. $86.5260$
Root an. cond. $9.30193$
Motivic weight $9$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−40.5 + 70.1i)3-s + (619. + 1.07e3i)5-s + (−1.33e3 − 6.21e3i)7-s + (−3.28e3 − 5.68e3i)9-s + (−242. + 420. i)11-s − 5.03e4·13-s − 1.00e5·15-s + (−4.94e4 + 8.56e4i)17-s + (1.35e5 + 2.34e5i)19-s + (4.89e5 + 1.57e5i)21-s + (−2.41e5 − 4.18e5i)23-s + (2.08e5 − 3.61e5i)25-s + 5.31e5·27-s + 3.37e6·29-s + (1.14e6 − 1.99e6i)31-s + ⋯
L(s)  = 1  + (−0.288 + 0.500i)3-s + (0.443 + 0.767i)5-s + (−0.210 − 0.977i)7-s + (−0.166 − 0.288i)9-s + (−0.00499 + 0.00865i)11-s − 0.488·13-s − 0.511·15-s + (−0.143 + 0.248i)17-s + (0.238 + 0.412i)19-s + (0.549 + 0.177i)21-s + (−0.180 − 0.311i)23-s + (0.106 − 0.185i)25-s + 0.192·27-s + 0.884·29-s + (0.223 − 0.387i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.366 - 0.930i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.366 - 0.930i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(168\)    =    \(2^{3} \cdot 3 \cdot 7\)
Sign: $0.366 - 0.930i$
Analytic conductor: \(86.5260\)
Root analytic conductor: \(9.30193\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{168} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 168,\ (\ :9/2),\ 0.366 - 0.930i)\)

Particular Values

\(L(5)\) \(\approx\) \(1.753196691\)
\(L(\frac12)\) \(\approx\) \(1.753196691\)
\(L(\frac{11}{2})\) 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 \( 1 + (40.5 - 70.1i)T \)
7 \( 1 + (1.33e3 + 6.21e3i)T \)
good5 \( 1 + (-619. - 1.07e3i)T + (-9.76e5 + 1.69e6i)T^{2} \)
11 \( 1 + (242. - 420. i)T + (-1.17e9 - 2.04e9i)T^{2} \)
13 \( 1 + 5.03e4T + 1.06e10T^{2} \)
17 \( 1 + (4.94e4 - 8.56e4i)T + (-5.92e10 - 1.02e11i)T^{2} \)
19 \( 1 + (-1.35e5 - 2.34e5i)T + (-1.61e11 + 2.79e11i)T^{2} \)
23 \( 1 + (2.41e5 + 4.18e5i)T + (-9.00e11 + 1.55e12i)T^{2} \)
29 \( 1 - 3.37e6T + 1.45e13T^{2} \)
31 \( 1 + (-1.14e6 + 1.99e6i)T + (-1.32e13 - 2.28e13i)T^{2} \)
37 \( 1 + (-4.97e6 - 8.61e6i)T + (-6.49e13 + 1.12e14i)T^{2} \)
41 \( 1 - 1.64e7T + 3.27e14T^{2} \)
43 \( 1 + 5.50e6T + 5.02e14T^{2} \)
47 \( 1 + (2.13e7 + 3.69e7i)T + (-5.59e14 + 9.69e14i)T^{2} \)
53 \( 1 + (8.72e6 - 1.51e7i)T + (-1.64e15 - 2.85e15i)T^{2} \)
59 \( 1 + (-1.92e7 + 3.34e7i)T + (-4.33e15 - 7.50e15i)T^{2} \)
61 \( 1 + (-4.36e7 - 7.56e7i)T + (-5.84e15 + 1.01e16i)T^{2} \)
67 \( 1 + (4.38e7 - 7.58e7i)T + (-1.36e16 - 2.35e16i)T^{2} \)
71 \( 1 - 2.43e8T + 4.58e16T^{2} \)
73 \( 1 + (5.75e7 - 9.96e7i)T + (-2.94e16 - 5.09e16i)T^{2} \)
79 \( 1 + (-2.30e8 - 3.99e8i)T + (-5.99e16 + 1.03e17i)T^{2} \)
83 \( 1 - 5.07e8T + 1.86e17T^{2} \)
89 \( 1 + (-5.49e7 - 9.51e7i)T + (-1.75e17 + 3.03e17i)T^{2} \)
97 \( 1 + 4.30e8T + 7.60e17T^{2} \)
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

−11.03962496175209950729124998357, −10.25104980595575006484068287044, −9.691254516971751541321997519161, −8.182721527936264260573473956244, −6.96243262845710346414794543118, −6.16368740774902661627674934288, −4.78773225643458813334247167867, −3.68778804998233743929752187324, −2.47803242407387159570811244404, −0.841306698814817148353836769650, 0.53760546808614036173761187958, 1.77258079126944907345658274734, 2.88330065991151610471059086320, 4.74937041627524625913239573693, 5.58211374445962523203445607207, 6.59770220207013251848894931480, 7.87697980433846587022183867246, 8.969547765338689097980335320051, 9.678036687350282333871088027503, 11.09117851590876725797104875874

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