Properties

Label 2-1850-1.1-c1-0-26
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 − 2·3-s + 4-s + 2·6-s − 1.37·7-s − 8-s + 9-s − 3.37·11-s − 2·12-s + 4.74·13-s + 1.37·14-s + 16-s + 5.37·17-s − 18-s − 2·19-s + 2.74·21-s + 3.37·22-s − 6.74·23-s + 2·24-s − 4.74·26-s + 4·27-s − 1.37·28-s + 8.11·29-s − 2.62·31-s − 32-s + 6.74·33-s − 5.37·34-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.15·3-s + 0.5·4-s + 0.816·6-s − 0.518·7-s − 0.353·8-s + 0.333·9-s − 1.01·11-s − 0.577·12-s + 1.31·13-s + 0.366·14-s + 0.250·16-s + 1.30·17-s − 0.235·18-s − 0.458·19-s + 0.598·21-s + 0.718·22-s − 1.40·23-s + 0.408·24-s − 0.930·26-s + 0.769·27-s − 0.259·28-s + 1.50·29-s − 0.471·31-s − 0.176·32-s + 1.17·33-s − 0.921·34-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: \(1\)
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 \( 1 + 2T + 3T^{2} \)
7 \( 1 + 1.37T + 7T^{2} \)
11 \( 1 + 3.37T + 11T^{2} \)
13 \( 1 - 4.74T + 13T^{2} \)
17 \( 1 - 5.37T + 17T^{2} \)
19 \( 1 + 2T + 19T^{2} \)
23 \( 1 + 6.74T + 23T^{2} \)
29 \( 1 - 8.11T + 29T^{2} \)
31 \( 1 + 2.62T + 31T^{2} \)
41 \( 1 - 5.37T + 41T^{2} \)
43 \( 1 + 7.37T + 43T^{2} \)
47 \( 1 - 8.74T + 47T^{2} \)
53 \( 1 + 1.37T + 53T^{2} \)
59 \( 1 - 12.7T + 59T^{2} \)
61 \( 1 + 5.37T + 61T^{2} \)
67 \( 1 + 4.74T + 67T^{2} \)
71 \( 1 + 6.74T + 71T^{2} \)
73 \( 1 + 8.74T + 73T^{2} \)
79 \( 1 + 4.74T + 79T^{2} \)
83 \( 1 - 0.744T + 83T^{2} \)
89 \( 1 - 10T + 89T^{2} \)
97 \( 1 - 0.116T + 97T^{2} \)
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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

−8.737052205505259617471811452516, −8.156267292721367562811560124814, −7.28981585347324473914742674662, −6.19036550881638327218872196326, −5.97709781165528597738580603587, −5.02561438281820363204533597119, −3.76187398807366899802760505357, −2.69766668476017202650903454300, −1.21018847376649232357277933766, 0, 1.21018847376649232357277933766, 2.69766668476017202650903454300, 3.76187398807366899802760505357, 5.02561438281820363204533597119, 5.97709781165528597738580603587, 6.19036550881638327218872196326, 7.28981585347324473914742674662, 8.156267292721367562811560124814, 8.737052205505259617471811452516

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