Properties

Label 2-392-49.15-c1-0-12
Degree $2$
Conductor $392$
Sign $-0.180 + 0.983i$
Analytic cond. $3.13013$
Root an. cond. $1.76921$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (1.84 − 2.31i)3-s + (0.101 − 0.127i)5-s + (−1.88 − 1.86i)7-s + (−1.28 − 5.61i)9-s + (−1.14 + 5.02i)11-s + (1.55 − 6.79i)13-s + (−0.107 − 0.471i)15-s + (1.76 − 0.848i)17-s − 2.11·19-s + (−7.77 + 0.918i)21-s + (6.97 + 3.35i)23-s + (1.10 + 4.84i)25-s + (−7.36 − 3.54i)27-s + (3.07 − 1.48i)29-s − 5.60·31-s + ⋯
L(s)  = 1  + (1.06 − 1.33i)3-s + (0.0455 − 0.0571i)5-s + (−0.710 − 0.703i)7-s + (−0.427 − 1.87i)9-s + (−0.345 + 1.51i)11-s + (0.430 − 1.88i)13-s + (−0.0277 − 0.121i)15-s + (0.427 − 0.205i)17-s − 0.485·19-s + (−1.69 + 0.200i)21-s + (1.45 + 0.700i)23-s + (0.221 + 0.969i)25-s + (−1.41 − 0.683i)27-s + (0.571 − 0.275i)29-s − 1.00·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.180 + 0.983i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.180 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(392\)    =    \(2^{3} \cdot 7^{2}\)
Sign: $-0.180 + 0.983i$
Analytic conductor: \(3.13013\)
Root analytic conductor: \(1.76921\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{392} (113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 392,\ (\ :1/2),\ -0.180 + 0.983i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.11927 - 1.34365i\)
\(L(\frac12)\) \(\approx\) \(1.11927 - 1.34365i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + (1.88 + 1.86i)T \)
good3 \( 1 + (-1.84 + 2.31i)T + (-0.667 - 2.92i)T^{2} \)
5 \( 1 + (-0.101 + 0.127i)T + (-1.11 - 4.87i)T^{2} \)
11 \( 1 + (1.14 - 5.02i)T + (-9.91 - 4.77i)T^{2} \)
13 \( 1 + (-1.55 + 6.79i)T + (-11.7 - 5.64i)T^{2} \)
17 \( 1 + (-1.76 + 0.848i)T + (10.5 - 13.2i)T^{2} \)
19 \( 1 + 2.11T + 19T^{2} \)
23 \( 1 + (-6.97 - 3.35i)T + (14.3 + 17.9i)T^{2} \)
29 \( 1 + (-3.07 + 1.48i)T + (18.0 - 22.6i)T^{2} \)
31 \( 1 + 5.60T + 31T^{2} \)
37 \( 1 + (2.50 - 1.20i)T + (23.0 - 28.9i)T^{2} \)
41 \( 1 + (-3.14 + 3.94i)T + (-9.12 - 39.9i)T^{2} \)
43 \( 1 + (-6.00 - 7.52i)T + (-9.56 + 41.9i)T^{2} \)
47 \( 1 + (-0.0480 + 0.210i)T + (-42.3 - 20.3i)T^{2} \)
53 \( 1 + (3.47 + 1.67i)T + (33.0 + 41.4i)T^{2} \)
59 \( 1 + (2.16 + 2.71i)T + (-13.1 + 57.5i)T^{2} \)
61 \( 1 + (-11.8 + 5.71i)T + (38.0 - 47.6i)T^{2} \)
67 \( 1 - 6.91T + 67T^{2} \)
71 \( 1 + (0.187 + 0.0901i)T + (44.2 + 55.5i)T^{2} \)
73 \( 1 + (-2.33 - 10.2i)T + (-65.7 + 31.6i)T^{2} \)
79 \( 1 + 7.91T + 79T^{2} \)
83 \( 1 + (0.979 + 4.28i)T + (-74.7 + 36.0i)T^{2} \)
89 \( 1 + (-1.35 - 5.92i)T + (-80.1 + 38.6i)T^{2} \)
97 \( 1 + 9.39T + 97T^{2} \)
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

−11.00650757772182279692565759231, −9.974178061410283180877374771235, −9.151077602921965809147682476831, −7.971185976155327771086402765910, −7.41425952250508232376030190078, −6.71608028173267713148971223544, −5.32582918736998756067043178532, −3.56992094796274763708230845665, −2.65629693473653674491060937521, −1.12713666738508434996960988806, 2.51809819931852625781782193582, 3.43647056274101362912174177164, 4.40832504568999370018915024472, 5.67625190580195705012953982536, 6.78694177676430871473250534917, 8.510440909858903149309181153310, 8.782311976950117629779145719027, 9.521846745656722847335007581096, 10.58968132627338175283725330293, 11.21214140442220630009247972539

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