Properties

Label 2-51e2-1.1-c1-0-68
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.765·2-s − 1.41·4-s + 0.0823·5-s + 2.76·7-s + 2.61·8-s − 0.0630·10-s − 0.668·11-s − 3.28·13-s − 2.11·14-s + 0.828·16-s + 3.64·19-s − 0.116·20-s + 0.511·22-s − 9.30·23-s − 4.99·25-s + 2.51·26-s − 3.91·28-s − 6.24·29-s + 5.04·31-s − 5.86·32-s + 0.227·35-s + 2.40·37-s − 2.78·38-s + 0.215·40-s + 0.480·41-s + 8.27·43-s + 0.944·44-s + ⋯
L(s)  = 1  − 0.541·2-s − 0.707·4-s + 0.0368·5-s + 1.04·7-s + 0.923·8-s − 0.0199·10-s − 0.201·11-s − 0.910·13-s − 0.565·14-s + 0.207·16-s + 0.835·19-s − 0.0260·20-s + 0.109·22-s − 1.94·23-s − 0.998·25-s + 0.492·26-s − 0.739·28-s − 1.15·29-s + 0.905·31-s − 1.03·32-s + 0.0385·35-s + 0.395·37-s − 0.451·38-s + 0.0340·40-s + 0.0749·41-s + 1.26·43-s + 0.142·44-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: \(1\)
Selberg data: \((2,\ 2601,\ (\ :1/2),\ -1)\)

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)$
bad3 \( 1 \)
17 \( 1 \)
good2 \( 1 + 0.765T + 2T^{2} \)
5 \( 1 - 0.0823T + 5T^{2} \)
7 \( 1 - 2.76T + 7T^{2} \)
11 \( 1 + 0.668T + 11T^{2} \)
13 \( 1 + 3.28T + 13T^{2} \)
19 \( 1 - 3.64T + 19T^{2} \)
23 \( 1 + 9.30T + 23T^{2} \)
29 \( 1 + 6.24T + 29T^{2} \)
31 \( 1 - 5.04T + 31T^{2} \)
37 \( 1 - 2.40T + 37T^{2} \)
41 \( 1 - 0.480T + 41T^{2} \)
43 \( 1 - 8.27T + 43T^{2} \)
47 \( 1 - 8.88T + 47T^{2} \)
53 \( 1 + 11.3T + 53T^{2} \)
59 \( 1 + 9.59T + 59T^{2} \)
61 \( 1 + 2.58T + 61T^{2} \)
67 \( 1 + 0.944T + 67T^{2} \)
71 \( 1 - 3.28T + 71T^{2} \)
73 \( 1 - 15.6T + 73T^{2} \)
79 \( 1 + 8.37T + 79T^{2} \)
83 \( 1 + 0.899T + 83T^{2} \)
89 \( 1 + 5.64T + 89T^{2} \)
97 \( 1 + 10.9T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.360263009944927757000147245480, −7.77130196546036899871603339612, −7.48761087766015719796858310140, −6.04603937300679044580523476420, −5.30605802880177524314827950275, −4.52685355196050944394682056597, −3.85226214424914784756853838359, −2.39165525280234694783411626298, −1.42200625665049204935669509806, 0, 1.42200625665049204935669509806, 2.39165525280234694783411626298, 3.85226214424914784756853838359, 4.52685355196050944394682056597, 5.30605802880177524314827950275, 6.04603937300679044580523476420, 7.48761087766015719796858310140, 7.77130196546036899871603339612, 8.360263009944927757000147245480

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