Properties

Label 2-8050-1.1-c1-0-148
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 + 4·11-s − 2·12-s − 14-s + 16-s − 6·17-s + 18-s − 6·19-s + 2·21-s + 4·22-s + 23-s − 2·24-s + 4·27-s − 28-s + 10·29-s + 4·31-s + 32-s − 8·33-s − 6·34-s + 36-s + 2·37-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 + 1.20·11-s − 0.577·12-s − 0.267·14-s + 1/4·16-s − 1.45·17-s + 0.235·18-s − 1.37·19-s + 0.436·21-s + 0.852·22-s + 0.208·23-s − 0.408·24-s + 0.769·27-s − 0.188·28-s + 1.85·29-s + 0.718·31-s + 0.176·32-s − 1.39·33-s − 1.02·34-s + 1/6·36-s + 0.328·37-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 \( 1 + 2 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 10 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 2 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 14 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
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

−6.88735318942094173624207569262, −6.53971592835965782705844673571, −6.33174311279454034272172298867, −5.39506292617571067794559020987, −4.52949482071829905188801506663, −4.29914013210746512672387954271, −3.19585646093781940426000197576, −2.30390877755402852348884298610, −1.20509634533530573283088519986, 0, 1.20509634533530573283088519986, 2.30390877755402852348884298610, 3.19585646093781940426000197576, 4.29914013210746512672387954271, 4.52949482071829905188801506663, 5.39506292617571067794559020987, 6.33174311279454034272172298867, 6.53971592835965782705844673571, 6.88735318942094173624207569262

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