Properties

Label 2-2075-1.1-c1-0-59
Degree $2$
Conductor $2075$
Sign $1$
Analytic cond. $16.5689$
Root an. cond. $4.07049$
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 + 3-s − 4-s + 6-s + 3·7-s − 3·8-s − 2·9-s + 3·11-s − 12-s + 6·13-s + 3·14-s − 16-s − 5·17-s − 2·18-s + 2·19-s + 3·21-s + 3·22-s + 4·23-s − 3·24-s + 6·26-s − 5·27-s − 3·28-s − 7·29-s + 5·31-s + 5·32-s + 3·33-s − 5·34-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s − 1/2·4-s + 0.408·6-s + 1.13·7-s − 1.06·8-s − 2/3·9-s + 0.904·11-s − 0.288·12-s + 1.66·13-s + 0.801·14-s − 1/4·16-s − 1.21·17-s − 0.471·18-s + 0.458·19-s + 0.654·21-s + 0.639·22-s + 0.834·23-s − 0.612·24-s + 1.17·26-s − 0.962·27-s − 0.566·28-s − 1.29·29-s + 0.898·31-s + 0.883·32-s + 0.522·33-s − 0.857·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2075 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2075 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2075\)    =    \(5^{2} \cdot 83\)
Sign: $1$
Analytic conductor: \(16.5689\)
Root analytic conductor: \(4.07049\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2075,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.018216739\)
\(L(\frac12)\) \(\approx\) \(3.018216739\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
83 \( 1 - T \)
good2 \( 1 - T + p T^{2} \)
3 \( 1 - T + p T^{2} \)
7 \( 1 - 3 T + p T^{2} \)
11 \( 1 - 3 T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 + 5 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 7 T + p T^{2} \)
31 \( 1 - 5 T + p T^{2} \)
37 \( 1 - 11 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 5 T + p T^{2} \)
61 \( 1 - 5 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 - 2 T + p T^{2} \)
73 \( 1 + p T^{2} \)
79 \( 1 - 14 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 - 8 T + p T^{2} \)
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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

−9.017571364405704341140281754032, −8.453475283691595638097668301696, −7.80720406011570304437105683036, −6.51003986895949489621827706154, −5.87263406714637014971627456813, −4.97148776811947809940547323320, −4.13105076781702502762968049773, −3.53133508802941266584945160071, −2.41615676868067838581180557878, −1.09189632316430757900963074575, 1.09189632316430757900963074575, 2.41615676868067838581180557878, 3.53133508802941266584945160071, 4.13105076781702502762968049773, 4.97148776811947809940547323320, 5.87263406714637014971627456813, 6.51003986895949489621827706154, 7.80720406011570304437105683036, 8.453475283691595638097668301696, 9.017571364405704341140281754032

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