↑

Properties

Label 2-392-56.5-c2-0-54
Degree $2$
Conductor $392$
Sign $-0.986 + 0.161i$
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  + (−1.33 − 1.48i)2-s + (−1.70 − 2.94i)3-s + (−0.432 + 3.97i)4-s + (2.15 − 3.73i)5-s + (−2.11 + 6.46i)6-s + (6.49 − 4.66i)8-s + (−1.28 + 2.23i)9-s + (−8.44 + 1.77i)10-s + (15.4 − 8.94i)11-s + (12.4 − 5.49i)12-s + 3.25·13-s − 14.6·15-s + (−15.6 − 3.43i)16-s + (13.6 − 7.86i)17-s + (5.04 − 1.06i)18-s + (0.778 − 1.34i)19-s + ⋯
L(s)  = 1  + (−0.667 − 0.744i)2-s + (−0.567 − 0.982i)3-s + (−0.108 + 0.994i)4-s + (0.431 − 0.747i)5-s + (−0.352 + 1.07i)6-s + (0.812 − 0.583i)8-s + (−0.143 + 0.248i)9-s + (−0.844 + 0.177i)10-s + (1.40 − 0.813i)11-s + (1.03 − 0.457i)12-s + 0.250·13-s − 0.979·15-s + (−0.976 − 0.214i)16-s + (0.801 − 0.462i)17-s + (0.280 − 0.0590i)18-s + (0.0409 − 0.0709i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.986 + 0.161i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.986 + 0.161i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(392\)    =    \(2^{3} \cdot 7^{2}\)
Sign: $-0.986 + 0.161i$
Analytic conductor: \(10.6812\)
Root analytic conductor: \(3.26821\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{392} (117, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 392,\ (\ :1),\ -0.986 + 0.161i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0920905 - 1.12973i\)
\(L(\frac12)\) \(\approx\) \(0.0920905 - 1.12973i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.33 + 1.48i)T \)
7 \( 1 \)
good3 \( 1 + (1.70 + 2.94i)T + (-4.5 + 7.79i)T^{2} \)
5 \( 1 + (-2.15 + 3.73i)T + (-12.5 - 21.6i)T^{2} \)
11 \( 1 + (-15.4 + 8.94i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 - 3.25T + 169T^{2} \)
17 \( 1 + (-13.6 + 7.86i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (-0.778 + 1.34i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (-20.7 + 35.8i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + 3.74iT - 841T^{2} \)
31 \( 1 + (-0.0145 + 0.00838i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (-1.16 - 0.674i)T + (684.5 + 1.18e3i)T^{2} \)
41 \( 1 - 70.3iT - 1.68e3T^{2} \)
43 \( 1 - 13.0iT - 1.84e3T^{2} \)
47 \( 1 + (30.9 + 17.8i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (39.7 - 22.9i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (34.3 + 59.4i)T + (-1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (48.0 - 83.2i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (12.0 - 6.97i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 75.7T + 5.04e3T^{2} \)
73 \( 1 + (-46.0 + 26.5i)T + (2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-11.6 + 20.2i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 102.T + 6.88e3T^{2} \)
89 \( 1 + (-76.6 - 44.2i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + 140. iT - 9.40e3T^{2} \)
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

−10.87509542262367172318630522002, −9.587853023915627954995335740862, −8.965588731786630124928634229058, −8.042050312279092113046734797900, −6.87813564703160362628140993861, −6.10182388005203006445022387719, −4.64119605598253547391721531118, −3.18781523940213048341490977843, −1.46234190279329743386538629291, −0.78816832903138081708546471417, 1.60599098520139249684201499292, 3.72184293425665944758543156182, 4.93303718601789803228698918009, 5.88671357202634357511063437001, 6.76327927319663490442477991098, 7.66509451700448555035689379039, 9.122843595603907216956837301070, 9.647876406591863882437197145004, 10.45338840008767115388600457619, 11.07838537270289165461897281227

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