Properties

Label 2-1850-1.1-c1-0-41
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 − 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: \(1\)
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 \( 1 + p T + p T^{2} \)
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

−9.006497938916409149052258052203, −7.65898904963165500718515206931, −6.90390686229064882950913520290, −6.41920364978516898967942841242, −5.39344798256389985367504415125, −4.99317541960821719676613529042, −4.22473947000803998945761355947, −2.91348169870031232108756983593, −1.51366604578487137578114595906, 0, 1.51366604578487137578114595906, 2.91348169870031232108756983593, 4.22473947000803998945761355947, 4.99317541960821719676613529042, 5.39344798256389985367504415125, 6.41920364978516898967942841242, 6.90390686229064882950913520290, 7.65898904963165500718515206931, 9.006497938916409149052258052203

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