Properties

Label 2-1850-1.1-c1-0-29
Degree $2$
Conductor $1850$
Sign $-1$
Analytic cond. $14.7723$
Root an. cond. $3.84347$
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 − 8-s + 9-s + 4·11-s − 2·12-s + 2·13-s + 16-s − 8·17-s − 18-s − 5·19-s − 4·22-s − 23-s + 2·24-s − 2·26-s + 4·27-s + 10·29-s − 4·31-s − 32-s − 8·33-s + 8·34-s + 36-s − 37-s + 5·38-s − 4·39-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.15·3-s + 1/2·4-s + 0.816·6-s − 0.353·8-s + 1/3·9-s + 1.20·11-s − 0.577·12-s + 0.554·13-s + 1/4·16-s − 1.94·17-s − 0.235·18-s − 1.14·19-s − 0.852·22-s − 0.208·23-s + 0.408·24-s − 0.392·26-s + 0.769·27-s + 1.85·29-s − 0.718·31-s − 0.176·32-s − 1.39·33-s + 1.37·34-s + 1/6·36-s − 0.164·37-s + 0.811·38-s − 0.640·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1850\)    =    \(2 \cdot 5^{2} \cdot 37\)
Sign: $-1$
Analytic conductor: \(14.7723\)
Root analytic conductor: \(3.84347\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1850} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1850,\ (\ :1/2),\ -1)\)

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 \( 1 \)
37 \( 1 + T \)
good3 \( 1 + 2 T + p T^{2} \)
7 \( 1 + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 8 T + p T^{2} \)
19 \( 1 + 5 T + p T^{2} \)
23 \( 1 + T + p T^{2} \)
29 \( 1 - 10 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
41 \( 1 - 7 T + p T^{2} \)
43 \( 1 - 9 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 - 3 T + p T^{2} \)
59 \( 1 + 11 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 - 14 T + p T^{2} \)
73 \( 1 + 3 T + p T^{2} \)
79 \( 1 + 11 T + p T^{2} \)
83 \( 1 + 8 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
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

−8.832914826819812321147395721742, −8.313688419694499349085941483242, −6.99952647899610143196985263278, −6.41576483049882713415932622776, −6.04441513309007745635467932400, −4.73812112133878014134329736510, −4.03316263506278759152895715932, −2.53069878250121404567366643784, −1.29600857288238140755146691958, 0, 1.29600857288238140755146691958, 2.53069878250121404567366643784, 4.03316263506278759152895715932, 4.73812112133878014134329736510, 6.04441513309007745635467932400, 6.41576483049882713415932622776, 6.99952647899610143196985263278, 8.313688419694499349085941483242, 8.832914826819812321147395721742

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