Properties

Label 2-4026-1.1-c1-0-81
Degree $2$
Conductor $4026$
Sign $-1$
Analytic cond. $32.1477$
Root an. cond. $5.66990$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3-s + 4-s + 1.77·5-s + 6-s + 2.51·7-s − 8-s + 9-s − 1.77·10-s + 11-s − 12-s − 6.37·13-s − 2.51·14-s − 1.77·15-s + 16-s + 0.125·17-s − 18-s + 1.93·19-s + 1.77·20-s − 2.51·21-s − 22-s + 6.49·23-s + 24-s − 1.84·25-s + 6.37·26-s − 27-s + 2.51·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.794·5-s + 0.408·6-s + 0.952·7-s − 0.353·8-s + 0.333·9-s − 0.562·10-s + 0.301·11-s − 0.288·12-s − 1.76·13-s − 0.673·14-s − 0.458·15-s + 0.250·16-s + 0.0303·17-s − 0.235·18-s + 0.443·19-s + 0.397·20-s − 0.549·21-s − 0.213·22-s + 1.35·23-s + 0.204·24-s − 0.368·25-s + 1.24·26-s − 0.192·27-s + 0.476·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4026\)    =    \(2 \cdot 3 \cdot 11 \cdot 61\)
Sign: $-1$
Analytic conductor: \(32.1477\)
Root analytic conductor: \(5.66990\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 4026,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 + T \)
11 \( 1 - T \)
61 \( 1 - T \)
good5 \( 1 - 1.77T + 5T^{2} \)
7 \( 1 - 2.51T + 7T^{2} \)
13 \( 1 + 6.37T + 13T^{2} \)
17 \( 1 - 0.125T + 17T^{2} \)
19 \( 1 - 1.93T + 19T^{2} \)
23 \( 1 - 6.49T + 23T^{2} \)
29 \( 1 + 9.05T + 29T^{2} \)
31 \( 1 + 5.91T + 31T^{2} \)
37 \( 1 + 4.87T + 37T^{2} \)
41 \( 1 + 4.27T + 41T^{2} \)
43 \( 1 - 9.43T + 43T^{2} \)
47 \( 1 + 13.0T + 47T^{2} \)
53 \( 1 + 0.558T + 53T^{2} \)
59 \( 1 + 3.30T + 59T^{2} \)
67 \( 1 + 10.9T + 67T^{2} \)
71 \( 1 + 11.6T + 71T^{2} \)
73 \( 1 + 0.0314T + 73T^{2} \)
79 \( 1 - 0.296T + 79T^{2} \)
83 \( 1 + 5.80T + 83T^{2} \)
89 \( 1 - 12.8T + 89T^{2} \)
97 \( 1 - 4.22T + 97T^{2} \)
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

−7.924790271245431722083477127092, −7.36910089922798617034705482516, −6.82593151950317453940452756993, −5.73792188723115105388368734538, −5.25261397508225639117185155802, −4.54119660902399518806844073757, −3.21136820300652684212357880163, −2.06164566843275293820219087746, −1.48536966765541131427338374957, 0, 1.48536966765541131427338374957, 2.06164566843275293820219087746, 3.21136820300652684212357880163, 4.54119660902399518806844073757, 5.25261397508225639117185155802, 5.73792188723115105388368734538, 6.82593151950317453940452756993, 7.36910089922798617034705482516, 7.924790271245431722083477127092

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