Properties

Label 2-51e2-1.1-c1-0-62
Degree $2$
Conductor $2601$
Sign $-1$
Analytic cond. $20.7690$
Root an. cond. $4.55731$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 4-s + 2·5-s − 7-s + 3·8-s − 2·10-s − 6·11-s + 13-s + 14-s − 16-s + 5·19-s − 2·20-s + 6·22-s + 2·23-s − 25-s − 26-s + 28-s + 6·29-s − 7·31-s − 5·32-s − 2·35-s − 7·37-s − 5·38-s + 6·40-s + 6·41-s + 7·43-s + 6·44-s + ⋯
L(s)  = 1  − 0.707·2-s − 1/2·4-s + 0.894·5-s − 0.377·7-s + 1.06·8-s − 0.632·10-s − 1.80·11-s + 0.277·13-s + 0.267·14-s − 1/4·16-s + 1.14·19-s − 0.447·20-s + 1.27·22-s + 0.417·23-s − 1/5·25-s − 0.196·26-s + 0.188·28-s + 1.11·29-s − 1.25·31-s − 0.883·32-s − 0.338·35-s − 1.15·37-s − 0.811·38-s + 0.948·40-s + 0.937·41-s + 1.06·43-s + 0.904·44-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: yes
Arithmetic: yes
Character: $\chi_{2601} (1, \cdot )$
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 \( 1 + T + p T^{2} \)
5 \( 1 - 2 T + p T^{2} \)
7 \( 1 + T + p T^{2} \)
11 \( 1 + 6 T + p T^{2} \)
13 \( 1 - T + p T^{2} \)
19 \( 1 - 5 T + p T^{2} \)
23 \( 1 - 2 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 7 T + p T^{2} \)
37 \( 1 + 7 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 7 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 - 12 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 + 11 T + p T^{2} \)
67 \( 1 + 11 T + p T^{2} \)
71 \( 1 + 2 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 2 T + p T^{2} \)
89 \( 1 - 16 T + p T^{2} \)
97 \( 1 + 17 T + p T^{2} \)
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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

−8.610669458588447357989850456492, −7.78521020073562122444728676100, −7.27333043666983284665548920512, −6.09334064493131882053312075649, −5.34125161449352351656292089800, −4.79216426991236760204650994159, −3.46988814885606600552792853543, −2.51976613306045938193861212532, −1.36741406968411664209091370409, 0, 1.36741406968411664209091370409, 2.51976613306045938193861212532, 3.46988814885606600552792853543, 4.79216426991236760204650994159, 5.34125161449352351656292089800, 6.09334064493131882053312075649, 7.27333043666983284665548920512, 7.78521020073562122444728676100, 8.610669458588447357989850456492

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