Properties

Label 2-51e2-1.1-c1-0-63
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  + 0.414·2-s − 1.82·4-s − 1.84·5-s + 1.08·7-s − 1.58·8-s − 0.765·10-s + 1.08·11-s − 1.41·13-s + 0.448·14-s + 3·16-s + 4.82·19-s + 3.37·20-s + 0.448·22-s + 4.14·23-s − 1.58·25-s − 0.585·26-s − 1.97·28-s − 4.46·29-s − 3.24·31-s + 4.41·32-s − 2·35-s + 3.82·37-s + 1.99·38-s + 2.93·40-s + 8.15·41-s − 4.82·43-s − 1.97·44-s + ⋯
L(s)  = 1  + 0.292·2-s − 0.914·4-s − 0.826·5-s + 0.409·7-s − 0.560·8-s − 0.242·10-s + 0.326·11-s − 0.392·13-s + 0.119·14-s + 0.750·16-s + 1.10·19-s + 0.755·20-s + 0.0955·22-s + 0.864·23-s − 0.317·25-s − 0.114·26-s − 0.374·28-s − 0.828·29-s − 0.583·31-s + 0.780·32-s − 0.338·35-s + 0.629·37-s + 0.324·38-s + 0.463·40-s + 1.27·41-s − 0.736·43-s − 0.298·44-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 \( 1 - 0.414T + 2T^{2} \)
5 \( 1 + 1.84T + 5T^{2} \)
7 \( 1 - 1.08T + 7T^{2} \)
11 \( 1 - 1.08T + 11T^{2} \)
13 \( 1 + 1.41T + 13T^{2} \)
19 \( 1 - 4.82T + 19T^{2} \)
23 \( 1 - 4.14T + 23T^{2} \)
29 \( 1 + 4.46T + 29T^{2} \)
31 \( 1 + 3.24T + 31T^{2} \)
37 \( 1 - 3.82T + 37T^{2} \)
41 \( 1 - 8.15T + 41T^{2} \)
43 \( 1 + 4.82T + 43T^{2} \)
47 \( 1 + 10.8T + 47T^{2} \)
53 \( 1 - 1.41T + 53T^{2} \)
59 \( 1 + 6T + 59T^{2} \)
61 \( 1 + 9.23T + 61T^{2} \)
67 \( 1 + 6.82T + 67T^{2} \)
71 \( 1 - 13.0T + 71T^{2} \)
73 \( 1 + 5.35T + 73T^{2} \)
79 \( 1 + 4.14T + 79T^{2} \)
83 \( 1 + 0.343T + 83T^{2} \)
89 \( 1 + 9.41T + 89T^{2} \)
97 \( 1 + 6.43T + 97T^{2} \)
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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

−8.401097205985173567790669928165, −7.80151761590282085877276898742, −7.15327102148086840762959053194, −6.00510496722109799378846393808, −5.17172453772659754687667631378, −4.53617257445345946304237262051, −3.73498416803677113178593549901, −2.98160547432116124720042936790, −1.36762499577204194674143885713, 0, 1.36762499577204194674143885713, 2.98160547432116124720042936790, 3.73498416803677113178593549901, 4.53617257445345946304237262051, 5.17172453772659754687667631378, 6.00510496722109799378846393808, 7.15327102148086840762959053194, 7.80151761590282085877276898742, 8.401097205985173567790669928165

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