Properties

Label 2-8050-1.1-c1-0-177
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 − 3·13-s − 14-s + 16-s − 17-s − 3·18-s + 4·22-s − 23-s − 3·26-s − 28-s − 4·31-s + 32-s − 34-s − 3·36-s + 11·37-s − 10·41-s − 2·43-s + 4·44-s − 46-s + 11·47-s + 49-s − 3·52-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 − 0.832·13-s − 0.267·14-s + 1/4·16-s − 0.242·17-s − 0.707·18-s + 0.852·22-s − 0.208·23-s − 0.588·26-s − 0.188·28-s − 0.718·31-s + 0.176·32-s − 0.171·34-s − 1/2·36-s + 1.80·37-s − 1.56·41-s − 0.304·43-s + 0.603·44-s − 0.147·46-s + 1.60·47-s + 1/7·49-s − 0.416·52-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 \( 1 + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 3 T + p T^{2} \)
17 \( 1 + T + p T^{2} \)
19 \( 1 + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 11 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 - 11 T + p T^{2} \)
53 \( 1 - T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 + 4 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 + 11 T + p T^{2} \)
83 \( 1 + 13 T + p T^{2} \)
89 \( 1 - T + p T^{2} \)
97 \( 1 + 7 T + p T^{2} \)
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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

−7.30614917040266229129834916137, −6.68940790572986483067064381183, −6.01202005800395902235385650023, −5.49724453957662851785330345356, −4.56679797372320399795364563211, −3.97031564998405657563814832929, −3.10310532824754280802888377210, −2.48195592761384522694417143693, −1.41998370484768490902893077241, 0, 1.41998370484768490902893077241, 2.48195592761384522694417143693, 3.10310532824754280802888377210, 3.97031564998405657563814832929, 4.56679797372320399795364563211, 5.49724453957662851785330345356, 6.01202005800395902235385650023, 6.68940790572986483067064381183, 7.30614917040266229129834916137

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