Properties

Label 2-2790-1.1-c1-0-19
Degree $2$
Conductor $2790$
Sign $1$
Analytic cond. $22.2782$
Root an. cond. $4.71998$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s − 5-s + 2·7-s + 8-s − 10-s + 4·11-s − 4·13-s + 2·14-s + 16-s + 6·17-s − 20-s + 4·22-s + 25-s − 4·26-s + 2·28-s − 4·29-s + 31-s + 32-s + 6·34-s − 2·35-s + 4·37-s − 40-s + 6·41-s − 8·43-s + 4·44-s + 12·47-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.755·7-s + 0.353·8-s − 0.316·10-s + 1.20·11-s − 1.10·13-s + 0.534·14-s + 1/4·16-s + 1.45·17-s − 0.223·20-s + 0.852·22-s + 1/5·25-s − 0.784·26-s + 0.377·28-s − 0.742·29-s + 0.179·31-s + 0.176·32-s + 1.02·34-s − 0.338·35-s + 0.657·37-s − 0.158·40-s + 0.937·41-s − 1.21·43-s + 0.603·44-s + 1.75·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2790 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2790 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2790\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 31\)
Sign: $1$
Analytic conductor: \(22.2782\)
Root analytic conductor: \(4.71998\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2790,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.185478637\)
\(L(\frac12)\) \(\approx\) \(3.185478637\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 + T \)
31 \( 1 - T \)
good7 \( 1 - 2 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 10 T + p T^{2} \)
71 \( 1 - 14 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 8 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−8.747416751953644846923547814896, −7.74335668534770922509830429662, −7.42070250982822790241988821325, −6.46093259456322875870891199741, −5.58739815873380500661416063334, −4.84972388474025854571339429019, −4.08822080273047061072314378148, −3.32306644259935075748578991415, −2.19359530048026004945282081289, −1.06631071912869019020178508767, 1.06631071912869019020178508767, 2.19359530048026004945282081289, 3.32306644259935075748578991415, 4.08822080273047061072314378148, 4.84972388474025854571339429019, 5.58739815873380500661416063334, 6.46093259456322875870891199741, 7.42070250982822790241988821325, 7.74335668534770922509830429662, 8.747416751953644846923547814896

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