Properties

Label 2-8050-1.1-c1-0-201
Degree $2$
Conductor $8050$
Sign $-1$
Analytic cond. $64.2795$
Root an. cond. $8.01745$
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 + 2·3-s + 4-s + 2·6-s − 7-s + 8-s + 9-s − 6·11-s + 2·12-s + 13-s − 14-s + 16-s + 3·17-s + 18-s − 4·19-s − 2·21-s − 6·22-s + 23-s + 2·24-s + 26-s − 4·27-s − 28-s − 6·29-s + 8·31-s + 32-s − 12·33-s + 3·34-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.15·3-s + 1/2·4-s + 0.816·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s − 1.80·11-s + 0.577·12-s + 0.277·13-s − 0.267·14-s + 1/4·16-s + 0.727·17-s + 0.235·18-s − 0.917·19-s − 0.436·21-s − 1.27·22-s + 0.208·23-s + 0.408·24-s + 0.196·26-s − 0.769·27-s − 0.188·28-s − 1.11·29-s + 1.43·31-s + 0.176·32-s − 2.08·33-s + 0.514·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8050\)    =    \(2 \cdot 5^{2} \cdot 7 \cdot 23\)
Sign: $-1$
Analytic conductor: \(64.2795\)
Root analytic conductor: \(8.01745\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 8050,\ (\ :1/2),\ -1)\)

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)$
bad2 \( 1 - T \)
5 \( 1 \)
7 \( 1 + T \)
23 \( 1 - T \)
good3 \( 1 - 2 T + p T^{2} \)
11 \( 1 + 6 T + p T^{2} \)
13 \( 1 - T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 11 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + 3 T + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + T + p T^{2} \)
83 \( 1 - 9 T + p T^{2} \)
89 \( 1 - 3 T + p T^{2} \)
97 \( 1 - T + p T^{2} \)
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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

−7.59604280605560956175024427547, −6.84220158127032164843302897128, −6.00212874467869519919736116645, −5.31734092371358798587742391545, −4.66086393172301257530890512148, −3.62087777678666329916006498037, −3.15510815113127334526718651560, −2.50007576998651018891514162695, −1.72376764281066899262318416372, 0, 1.72376764281066899262318416372, 2.50007576998651018891514162695, 3.15510815113127334526718651560, 3.62087777678666329916006498037, 4.66086393172301257530890512148, 5.31734092371358798587742391545, 6.00212874467869519919736116645, 6.84220158127032164843302897128, 7.59604280605560956175024427547

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