Properties

Label 2-1850-1.1-c1-0-18
Degree $2$
Conductor $1850$
Sign $1$
Analytic cond. $14.7723$
Root an. cond. $3.84347$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 1.09·3-s + 4-s − 1.09·6-s + 3.20·7-s − 8-s − 1.80·9-s + 3.82·11-s + 1.09·12-s − 0.147·13-s − 3.20·14-s + 16-s − 0.978·17-s + 1.80·18-s + 2.67·19-s + 3.51·21-s − 3.82·22-s + 2.33·23-s − 1.09·24-s + 0.147·26-s − 5.25·27-s + 3.20·28-s + 6.30·29-s − 3.62·31-s − 32-s + 4.18·33-s + 0.978·34-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.631·3-s + 0.5·4-s − 0.446·6-s + 1.21·7-s − 0.353·8-s − 0.600·9-s + 1.15·11-s + 0.315·12-s − 0.0408·13-s − 0.857·14-s + 0.250·16-s − 0.237·17-s + 0.424·18-s + 0.613·19-s + 0.766·21-s − 0.815·22-s + 0.487·23-s − 0.223·24-s + 0.0288·26-s − 1.01·27-s + 0.606·28-s + 1.17·29-s − 0.650·31-s − 0.176·32-s + 0.728·33-s + 0.167·34-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: no
Arithmetic: yes
Character: $\chi_{1850} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1850,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.874617049\)
\(L(\frac12)\) \(\approx\) \(1.874617049\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.09T + 3T^{2} \)
7 \( 1 - 3.20T + 7T^{2} \)
11 \( 1 - 3.82T + 11T^{2} \)
13 \( 1 + 0.147T + 13T^{2} \)
17 \( 1 + 0.978T + 17T^{2} \)
19 \( 1 - 2.67T + 19T^{2} \)
23 \( 1 - 2.33T + 23T^{2} \)
29 \( 1 - 6.30T + 29T^{2} \)
31 \( 1 + 3.62T + 31T^{2} \)
41 \( 1 - 11.8T + 41T^{2} \)
43 \( 1 - 4.53T + 43T^{2} \)
47 \( 1 + 6.23T + 47T^{2} \)
53 \( 1 + 11.2T + 53T^{2} \)
59 \( 1 - 6.92T + 59T^{2} \)
61 \( 1 - 10.4T + 61T^{2} \)
67 \( 1 + 2.80T + 67T^{2} \)
71 \( 1 + 12.3T + 71T^{2} \)
73 \( 1 - 13.9T + 73T^{2} \)
79 \( 1 - 15.6T + 79T^{2} \)
83 \( 1 + 13.5T + 83T^{2} \)
89 \( 1 + 6.46T + 89T^{2} \)
97 \( 1 - 3.07T + 97T^{2} \)
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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

−9.149632890408981723829804584040, −8.474766968965112134627029884307, −7.919864124275438965811064285926, −7.12060596099840916406405446605, −6.19492570500117007327145885226, −5.21555171778293336272398856608, −4.20008876152689217013182376358, −3.12615709160101338142673300366, −2.10666321650458873868823471400, −1.07584621765779447049463277129, 1.07584621765779447049463277129, 2.10666321650458873868823471400, 3.12615709160101338142673300366, 4.20008876152689217013182376358, 5.21555171778293336272398856608, 6.19492570500117007327145885226, 7.12060596099840916406405446605, 7.919864124275438965811064285926, 8.474766968965112134627029884307, 9.149632890408981723829804584040

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