Properties

Label 2-8050-1.1-c1-0-171
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 + 4-s − 7-s + 8-s − 3·9-s − 4·11-s + 13-s − 14-s + 16-s + 3·17-s − 3·18-s + 8·19-s − 4·22-s − 23-s + 26-s − 28-s − 8·29-s + 4·31-s + 32-s + 3·34-s − 3·36-s − 5·37-s + 8·38-s + 6·41-s − 2·43-s − 4·44-s − 46-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 0.377·7-s + 0.353·8-s − 9-s − 1.20·11-s + 0.277·13-s − 0.267·14-s + 1/4·16-s + 0.727·17-s − 0.707·18-s + 1.83·19-s − 0.852·22-s − 0.208·23-s + 0.196·26-s − 0.188·28-s − 1.48·29-s + 0.718·31-s + 0.176·32-s + 0.514·34-s − 1/2·36-s − 0.821·37-s + 1.29·38-s + 0.937·41-s − 0.304·43-s − 0.603·44-s − 0.147·46-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: $\chi_{8050} (1, \cdot )$
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 \( 1 + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 - T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 5 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 + T + p T^{2} \)
53 \( 1 + 7 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 - 5 T + p T^{2} \)
83 \( 1 + 17 T + p T^{2} \)
89 \( 1 + 3 T + p T^{2} \)
97 \( 1 - 13 T + p T^{2} \)
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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

−7.67464733429060645778310131267, −6.69872776615220740694218802442, −5.73728690380499158817190403585, −5.56193316633919703115422799526, −4.83876383606479452472076041801, −3.73559610665724590884101397729, −3.10338747966418723612624547974, −2.59745316070575425828374178212, −1.36643612662942117847805433889, 0, 1.36643612662942117847805433889, 2.59745316070575425828374178212, 3.10338747966418723612624547974, 3.73559610665724590884101397729, 4.83876383606479452472076041801, 5.56193316633919703115422799526, 5.73728690380499158817190403585, 6.69872776615220740694218802442, 7.67464733429060645778310131267

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