↑

Properties

Label 2-392-56.11-c2-0-62
Degree $2$
Conductor $392$
Sign $0.907 + 0.420i$
Analytic cond. $10.6812$
Root an. cond. $3.26821$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (1.97 − 0.323i)2-s + (−0.292 + 0.507i)3-s + (3.79 − 1.27i)4-s + (7.82 − 4.51i)5-s + (−0.414 + 1.09i)6-s + (7.07 − 3.74i)8-s + (4.32 + 7.49i)9-s + (13.9 − 11.4i)10-s + (−6.24 + 10.8i)11-s + (−0.463 + 2.29i)12-s − 9.03i·13-s + 5.29i·15-s + (12.7 − 9.66i)16-s + (−6.17 + 10.6i)17-s + (10.9 + 13.3i)18-s + (−14.4 − 25.0i)19-s + ⋯
L(s)  = 1  + (0.986 − 0.161i)2-s + (−0.0976 + 0.169i)3-s + (0.947 − 0.318i)4-s + (1.56 − 0.903i)5-s + (−0.0690 + 0.182i)6-s + (0.883 − 0.467i)8-s + (0.480 + 0.833i)9-s + (1.39 − 1.14i)10-s + (−0.567 + 0.982i)11-s + (−0.0386 + 0.191i)12-s − 0.694i·13-s + 0.352i·15-s + (0.796 − 0.604i)16-s + (−0.363 + 0.628i)17-s + (0.609 + 0.744i)18-s + (−0.759 − 1.31i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.907 + 0.420i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.907 + 0.420i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(392\)    =    \(2^{3} \cdot 7^{2}\)
Sign: $0.907 + 0.420i$
Analytic conductor: \(10.6812\)
Root analytic conductor: \(3.26821\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{392} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 392,\ (\ :1),\ 0.907 + 0.420i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(3.80764 - 0.838982i\)
\(L(\frac12)\) \(\approx\) \(3.80764 - 0.838982i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.97 + 0.323i)T \)
7 \( 1 \)
good3 \( 1 + (0.292 - 0.507i)T + (-4.5 - 7.79i)T^{2} \)
5 \( 1 + (-7.82 + 4.51i)T + (12.5 - 21.6i)T^{2} \)
11 \( 1 + (6.24 - 10.8i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 + 9.03iT - 169T^{2} \)
17 \( 1 + (6.17 - 10.6i)T + (-144.5 - 250. i)T^{2} \)
19 \( 1 + (14.4 + 25.0i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (21.3 - 12.3i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 - 22.4iT - 841T^{2} \)
31 \( 1 + (-14.5 - 8.39i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (-14.0 + 8.12i)T + (684.5 - 1.18e3i)T^{2} \)
41 \( 1 - 6.97T + 1.68e3T^{2} \)
43 \( 1 + 22.8T + 1.84e3T^{2} \)
47 \( 1 + (5.36 - 3.09i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (6.94 + 4.00i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (15.2 - 26.3i)T + (-1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (13.1 - 7.61i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-39.3 + 68.0i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 17.5iT - 5.04e3T^{2} \)
73 \( 1 + (23.3 - 40.4i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (70.1 - 40.5i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 40.3T + 6.88e3T^{2} \)
89 \( 1 + (55.9 + 96.9i)T + (-3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + 164.T + 9.40e3T^{2} \)
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

−10.86478981927509311346080021369, −10.26528089843198422511533432546, −9.534010827534152446672176545179, −8.211332270771191031469817820567, −6.96926736250280550825107190048, −5.88537374527757210818954265064, −5.06469189467754042557606289430, −4.45455125392897318798270224884, −2.49116061879792985742416930648, −1.65164328733144217688609638787, 1.84509032184450168853680816366, 2.88507919816414874720348957065, 4.16613170307746630643215470744, 5.68040283338879637568583427055, 6.24283485659344221628453018622, 6.86927648801801665280233922600, 8.196935699677977431616552487075, 9.623344721612198799117583719461, 10.33738534994051994278790486855, 11.22200986828105581692139292461

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