Properties

Label 2-1850-1.1-c1-0-27
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3·3-s + 4-s − 3·6-s − 8-s + 6·9-s − 11-s + 3·12-s + 2·13-s + 16-s + 7·17-s − 6·18-s + 5·19-s + 22-s − 6·23-s − 3·24-s − 2·26-s + 9·27-s − 4·31-s − 32-s − 3·33-s − 7·34-s + 6·36-s − 37-s − 5·38-s + 6·39-s − 3·41-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.73·3-s + 1/2·4-s − 1.22·6-s − 0.353·8-s + 2·9-s − 0.301·11-s + 0.866·12-s + 0.554·13-s + 1/4·16-s + 1.69·17-s − 1.41·18-s + 1.14·19-s + 0.213·22-s − 1.25·23-s − 0.612·24-s − 0.392·26-s + 1.73·27-s − 0.718·31-s − 0.176·32-s − 0.522·33-s − 1.20·34-s + 36-s − 0.164·37-s − 0.811·38-s + 0.960·39-s − 0.468·41-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: \(0\)
Selberg data: \((2,\ 1850,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.558578453\)
\(L(\frac12)\) \(\approx\) \(2.558578453\)
\(L(\frac{3}{2})\) 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 \( 1 - p T + p T^{2} \)
7 \( 1 + p T^{2} \)
11 \( 1 + T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 7 T + p T^{2} \)
19 \( 1 - 5 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
41 \( 1 + 3 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 - 13 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 - 7 T + p T^{2} \)
79 \( 1 - 14 T + p T^{2} \)
83 \( 1 + 3 T + p T^{2} \)
89 \( 1 + 7 T + p T^{2} \)
97 \( 1 - 18 T + p T^{2} \)
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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

−9.177649149000980779645265803026, −8.459325576811713568506936791912, −7.68736201597947535128658618989, −7.53300371838530950412027477632, −6.25262510775209233600867082738, −5.20732494807423981373300369721, −3.74965680760340825728557237450, −3.27661932383606895541022078526, −2.23400282973532377904024408291, −1.23504527578568251780025983879, 1.23504527578568251780025983879, 2.23400282973532377904024408291, 3.27661932383606895541022078526, 3.74965680760340825728557237450, 5.20732494807423981373300369721, 6.25262510775209233600867082738, 7.53300371838530950412027477632, 7.68736201597947535128658618989, 8.459325576811713568506936791912, 9.177649149000980779645265803026

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