Properties

Label 2-1850-1.1-c1-0-47
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 1.44·3-s + 4-s − 1.44·6-s − 2.44·7-s − 8-s − 0.898·9-s + 3.44·11-s + 1.44·12-s + 0.449·13-s + 2.44·14-s + 16-s − 3.44·17-s + 0.898·18-s − 5·19-s − 3.55·21-s − 3.44·22-s − 2·23-s − 1.44·24-s − 0.449·26-s − 5.65·27-s − 2.44·28-s − 0.898·29-s + 4.44·31-s − 32-s + 5·33-s + 3.44·34-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.836·3-s + 0.5·4-s − 0.591·6-s − 0.925·7-s − 0.353·8-s − 0.299·9-s + 1.04·11-s + 0.418·12-s + 0.124·13-s + 0.654·14-s + 0.250·16-s − 0.836·17-s + 0.211·18-s − 1.14·19-s − 0.774·21-s − 0.735·22-s − 0.417·23-s − 0.295·24-s − 0.0881·26-s − 1.08·27-s − 0.462·28-s − 0.166·29-s + 0.799·31-s − 0.176·32-s + 0.870·33-s + 0.591·34-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: Trivial
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 \( 1 - 1.44T + 3T^{2} \)
7 \( 1 + 2.44T + 7T^{2} \)
11 \( 1 - 3.44T + 11T^{2} \)
13 \( 1 - 0.449T + 13T^{2} \)
17 \( 1 + 3.44T + 17T^{2} \)
19 \( 1 + 5T + 19T^{2} \)
23 \( 1 + 2T + 23T^{2} \)
29 \( 1 + 0.898T + 29T^{2} \)
31 \( 1 - 4.44T + 31T^{2} \)
41 \( 1 - T + 41T^{2} \)
43 \( 1 - 1.10T + 43T^{2} \)
47 \( 1 + 9.79T + 47T^{2} \)
53 \( 1 + 6T + 53T^{2} \)
59 \( 1 + 2T + 59T^{2} \)
61 \( 1 + 6.44T + 61T^{2} \)
67 \( 1 + 4.55T + 67T^{2} \)
71 \( 1 + 7.55T + 71T^{2} \)
73 \( 1 + 12.7T + 73T^{2} \)
79 \( 1 - 7.79T + 79T^{2} \)
83 \( 1 + 3.44T + 83T^{2} \)
89 \( 1 - 14.3T + 89T^{2} \)
97 \( 1 + 14T + 97T^{2} \)
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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

−8.864720741768348609016183474117, −8.334834854918181383278511702911, −7.42662619777200227219034265936, −6.39901597490605843398317361077, −6.15592620182016360245135055390, −4.51399061144460739509022220272, −3.56290718360916960598390387838, −2.73885823713996759335050505992, −1.72506485647856065953250782080, 0, 1.72506485647856065953250782080, 2.73885823713996759335050505992, 3.56290718360916960598390387838, 4.51399061144460739509022220272, 6.15592620182016360245135055390, 6.39901597490605843398317361077, 7.42662619777200227219034265936, 8.334834854918181383278511702911, 8.864720741768348609016183474117

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