↑

Properties

Label 2-336-7.4-c5-0-10
Degree $2$
Conductor $336$
Sign $0.898 - 0.439i$
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 + (−7.86 − 13.6i)5-s + (−129. − 0.644i)7-s + (−40.5 − 70.1i)9-s + (−34.0 + 58.9i)11-s − 506.·13-s − 141.·15-s + (−92.4 + 160. i)17-s + (671. + 1.16e3i)19-s + (−588. + 1.00e3i)21-s + (−85.7 − 148. i)23-s + (1.43e3 − 2.49e3i)25-s − 729·27-s + 4.74e3·29-s + (−3.70e3 + 6.41e3i)31-s + ⋯
L(s)  = 1  + (0.288 − 0.499i)3-s + (−0.140 − 0.243i)5-s + (−0.999 − 0.00497i)7-s + (−0.166 − 0.288i)9-s + (−0.0847 + 0.146i)11-s − 0.831·13-s − 0.162·15-s + (−0.0775 + 0.134i)17-s + (0.426 + 0.738i)19-s + (−0.291 + 0.498i)21-s + (−0.0337 − 0.0585i)23-s + (0.460 − 0.797i)25-s − 0.192·27-s + 1.04·29-s + (−0.692 + 1.19i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.898 - 0.439i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.898 - 0.439i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.898 - 0.439i$
Analytic conductor: \(53.8889\)
Root analytic conductor: \(7.34091\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :5/2),\ 0.898 - 0.439i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.416714234\)
\(L(\frac12)\) \(\approx\) \(1.416714234\)
\(L(\frac{7}{2})\) 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 \( 1 + (129. + 0.644i)T \)
good5 \( 1 + (7.86 + 13.6i)T + (-1.56e3 + 2.70e3i)T^{2} \)
11 \( 1 + (34.0 - 58.9i)T + (-8.05e4 - 1.39e5i)T^{2} \)
13 \( 1 + 506.T + 3.71e5T^{2} \)
17 \( 1 + (92.4 - 160. i)T + (-7.09e5 - 1.22e6i)T^{2} \)
19 \( 1 + (-671. - 1.16e3i)T + (-1.23e6 + 2.14e6i)T^{2} \)
23 \( 1 + (85.7 + 148. i)T + (-3.21e6 + 5.57e6i)T^{2} \)
29 \( 1 - 4.74e3T + 2.05e7T^{2} \)
31 \( 1 + (3.70e3 - 6.41e3i)T + (-1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 + (-2.25e3 - 3.91e3i)T + (-3.46e7 + 6.00e7i)T^{2} \)
41 \( 1 - 1.39e4T + 1.15e8T^{2} \)
43 \( 1 + 3.31e3T + 1.47e8T^{2} \)
47 \( 1 + (-256. - 444. i)T + (-1.14e8 + 1.98e8i)T^{2} \)
53 \( 1 + (-5.62e3 + 9.74e3i)T + (-2.09e8 - 3.62e8i)T^{2} \)
59 \( 1 + (1.32e4 - 2.29e4i)T + (-3.57e8 - 6.19e8i)T^{2} \)
61 \( 1 + (2.85e3 + 4.94e3i)T + (-4.22e8 + 7.31e8i)T^{2} \)
67 \( 1 + (8.36e3 - 1.44e4i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 + 1.01e4T + 1.80e9T^{2} \)
73 \( 1 + (-3.72e4 + 6.45e4i)T + (-1.03e9 - 1.79e9i)T^{2} \)
79 \( 1 + (-1.86e4 - 3.23e4i)T + (-1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 - 8.92e4T + 3.93e9T^{2} \)
89 \( 1 + (-4.81e4 - 8.33e4i)T + (-2.79e9 + 4.83e9i)T^{2} \)
97 \( 1 - 8.10e3T + 8.58e9T^{2} \)
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

−10.64792188885861613645377613011, −9.808733974547134967343225058145, −8.922993168613461475916893297869, −7.911018359362797135044307842115, −6.97649256305553326330162459420, −6.09508829194504907708569409121, −4.80072138212909872840145535150, −3.46036577071789401840939402401, −2.40155657259022171725011750869, −0.874114815452430386794279854966, 0.45504007506216853514173287993, 2.47801104312019832426119565143, 3.35613315545274450387140156344, 4.55069566642583364229503288255, 5.71943297961533413909940134501, 6.89159729514271208366053547120, 7.74335329705190727596123762189, 9.091787119298215092000383833559, 9.603426946786299878901409001459, 10.57100635813569167693149923211

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