Properties

Label 2-51e2-1.1-c1-0-23
Degree $2$
Conductor $2601$
Sign $1$
Analytic cond. $20.7690$
Root an. cond. $4.55731$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2.14·2-s + 2.58·4-s + 2.79·5-s − 2.93·7-s − 1.25·8-s − 5.99·10-s − 1.15·11-s + 4.41·13-s + 6.27·14-s − 2.48·16-s + 6.41·19-s + 7.23·20-s + 2.48·22-s − 2.79·23-s + 2.82·25-s − 9.45·26-s − 7.57·28-s − 9.55·29-s + 7.83·31-s + 7.83·32-s − 8.19·35-s − 1.39·37-s − 13.7·38-s − 3.50·40-s + 0.480·41-s + 8.07·43-s − 2.99·44-s + ⋯
L(s)  = 1  − 1.51·2-s + 1.29·4-s + 1.25·5-s − 1.10·7-s − 0.443·8-s − 1.89·10-s − 0.349·11-s + 1.22·13-s + 1.67·14-s − 0.621·16-s + 1.47·19-s + 1.61·20-s + 0.529·22-s − 0.583·23-s + 0.565·25-s − 1.85·26-s − 1.43·28-s − 1.77·29-s + 1.40·31-s + 1.38·32-s − 1.38·35-s − 0.230·37-s − 2.22·38-s − 0.554·40-s + 0.0749·41-s + 1.23·43-s − 0.451·44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2601 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2601 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2601\)    =    \(3^{2} \cdot 17^{2}\)
Sign: $1$
Analytic conductor: \(20.7690\)
Root analytic conductor: \(4.55731\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2601,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9492649336\)
\(L(\frac12)\) \(\approx\) \(0.9492649336\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
17 \( 1 \)
good2 \( 1 + 2.14T + 2T^{2} \)
5 \( 1 - 2.79T + 5T^{2} \)
7 \( 1 + 2.93T + 7T^{2} \)
11 \( 1 + 1.15T + 11T^{2} \)
13 \( 1 - 4.41T + 13T^{2} \)
19 \( 1 - 6.41T + 19T^{2} \)
23 \( 1 + 2.79T + 23T^{2} \)
29 \( 1 + 9.55T + 29T^{2} \)
31 \( 1 - 7.83T + 31T^{2} \)
37 \( 1 + 1.39T + 37T^{2} \)
41 \( 1 - 0.480T + 41T^{2} \)
43 \( 1 - 8.07T + 43T^{2} \)
47 \( 1 - 6.05T + 47T^{2} \)
53 \( 1 + 10.3T + 53T^{2} \)
59 \( 1 - 6.42T + 59T^{2} \)
61 \( 1 - 0.765T + 61T^{2} \)
67 \( 1 - 0.585T + 67T^{2} \)
71 \( 1 - 12.8T + 71T^{2} \)
73 \( 1 - 11.9T + 73T^{2} \)
79 \( 1 - 2.48T + 79T^{2} \)
83 \( 1 + 10.3T + 83T^{2} \)
89 \( 1 - 11.2T + 89T^{2} \)
97 \( 1 + 6.30T + 97T^{2} \)
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

−9.199447490067227938996482103656, −8.248220923923704536340301758283, −7.54341847450070772339922513962, −6.63899554714074536325482729866, −6.05688280835954651144371407434, −5.33359311573111340995107102276, −3.83064456385572029254802696730, −2.78192789151636231912830142056, −1.81500707767230118301835283509, −0.78751591495338882540573691967, 0.78751591495338882540573691967, 1.81500707767230118301835283509, 2.78192789151636231912830142056, 3.83064456385572029254802696730, 5.33359311573111340995107102276, 6.05688280835954651144371407434, 6.63899554714074536325482729866, 7.54341847450070772339922513962, 8.248220923923704536340301758283, 9.199447490067227938996482103656

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