↑

Properties

Label 2-336-7.2-c5-0-15
Degree $2$
Conductor $336$
Sign $0.890 - 0.454i$
Analytic cond. $53.8889$
Root an. cond. $7.34091$
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  + (−4.5 − 7.79i)3-s + (8.04 − 13.9i)5-s + (77.3 − 104. i)7-s + (−40.5 + 70.1i)9-s + (329. + 570. i)11-s + 196.·13-s − 144.·15-s + (−228. − 395. i)17-s + (−517. + 896. i)19-s + (−1.15e3 − 134. i)21-s + (−1.12e3 + 1.94e3i)23-s + (1.43e3 + 2.48e3i)25-s + 729·27-s − 8.25e3·29-s + (2.89e3 + 5.02e3i)31-s + ⋯
L(s)  = 1  + (−0.288 − 0.499i)3-s + (0.143 − 0.249i)5-s + (0.596 − 0.802i)7-s + (−0.166 + 0.288i)9-s + (0.820 + 1.42i)11-s + 0.323·13-s − 0.166·15-s + (−0.191 − 0.332i)17-s + (−0.329 + 0.569i)19-s + (−0.573 − 0.0666i)21-s + (−0.442 + 0.766i)23-s + (0.458 + 0.794i)25-s + 0.192·27-s − 1.82·29-s + (0.541 + 0.938i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.890 - 0.454i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.890 - 0.454i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.890 - 0.454i$
Analytic conductor: \(53.8889\)
Root analytic conductor: \(7.34091\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :5/2),\ 0.890 - 0.454i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.902439450\)
\(L(\frac12)\) \(\approx\) \(1.902439450\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (4.5 + 7.79i)T \)
7 \( 1 + (-77.3 + 104. i)T \)
good5 \( 1 + (-8.04 + 13.9i)T + (-1.56e3 - 2.70e3i)T^{2} \)
11 \( 1 + (-329. - 570. i)T + (-8.05e4 + 1.39e5i)T^{2} \)
13 \( 1 - 196.T + 3.71e5T^{2} \)
17 \( 1 + (228. + 395. i)T + (-7.09e5 + 1.22e6i)T^{2} \)
19 \( 1 + (517. - 896. i)T + (-1.23e6 - 2.14e6i)T^{2} \)
23 \( 1 + (1.12e3 - 1.94e3i)T + (-3.21e6 - 5.57e6i)T^{2} \)
29 \( 1 + 8.25e3T + 2.05e7T^{2} \)
31 \( 1 + (-2.89e3 - 5.02e3i)T + (-1.43e7 + 2.47e7i)T^{2} \)
37 \( 1 + (-4.15e3 + 7.19e3i)T + (-3.46e7 - 6.00e7i)T^{2} \)
41 \( 1 - 1.45e4T + 1.15e8T^{2} \)
43 \( 1 - 796.T + 1.47e8T^{2} \)
47 \( 1 + (1.10e4 - 1.91e4i)T + (-1.14e8 - 1.98e8i)T^{2} \)
53 \( 1 + (7.06e3 + 1.22e4i)T + (-2.09e8 + 3.62e8i)T^{2} \)
59 \( 1 + (-1.76e4 - 3.05e4i)T + (-3.57e8 + 6.19e8i)T^{2} \)
61 \( 1 + (1.57e4 - 2.73e4i)T + (-4.22e8 - 7.31e8i)T^{2} \)
67 \( 1 + (-2.48e4 - 4.30e4i)T + (-6.75e8 + 1.16e9i)T^{2} \)
71 \( 1 - 7.90e3T + 1.80e9T^{2} \)
73 \( 1 + (8.02e3 + 1.38e4i)T + (-1.03e9 + 1.79e9i)T^{2} \)
79 \( 1 + (-9.81e3 + 1.70e4i)T + (-1.53e9 - 2.66e9i)T^{2} \)
83 \( 1 - 6.04e4T + 3.93e9T^{2} \)
89 \( 1 + (-1.91e4 + 3.31e4i)T + (-2.79e9 - 4.83e9i)T^{2} \)
97 \( 1 - 5.72e4T + 8.58e9T^{2} \)
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

−10.94250613174108154649532692674, −9.852954614558065803388333142667, −8.985174877303683203319973951483, −7.64554338629857704506837735418, −7.16075319872750278074397541684, −5.96406108408523409678480244025, −4.78086538586664486438323989284, −3.83785907688978812703037332915, −1.93564172692035738380782720774, −1.12604923028314657449960771338, 0.57767608504449574929572480517, 2.19135512409305431633079437666, 3.52267938515870214381166411833, 4.65439820509835351004655822571, 5.89289585170776570808597873174, 6.41573829623271787912682959843, 8.089635854561184054595320371584, 8.794627988717365554488168417536, 9.656040597197112665503313600782, 11.02304495341215551074803463908

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