Properties

Label 2-1850-1.1-c1-0-3
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 − 2.72·3-s + 4-s − 2.72·6-s − 4.14·7-s + 8-s + 4.45·9-s − 4.76·11-s − 2.72·12-s − 3.91·13-s − 4.14·14-s + 16-s + 3.31·17-s + 4.45·18-s − 1.85·19-s + 11.3·21-s − 4.76·22-s − 1.54·23-s − 2.72·24-s − 3.91·26-s − 3.96·27-s − 4.14·28-s + 8.87·29-s + 9.75·31-s + 32-s + 13.0·33-s + 3.31·34-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.57·3-s + 0.5·4-s − 1.11·6-s − 1.56·7-s + 0.353·8-s + 1.48·9-s − 1.43·11-s − 0.788·12-s − 1.08·13-s − 1.10·14-s + 0.250·16-s + 0.804·17-s + 1.04·18-s − 0.424·19-s + 2.46·21-s − 1.01·22-s − 0.321·23-s − 0.557·24-s − 0.768·26-s − 0.762·27-s − 0.783·28-s + 1.64·29-s + 1.75·31-s + 0.176·32-s + 2.26·33-s + 0.568·34-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

\(L(1)\) \(\approx\) \(0.7970439972\)
\(L(\frac12)\) \(\approx\) \(0.7970439972\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.72T + 3T^{2} \)
7 \( 1 + 4.14T + 7T^{2} \)
11 \( 1 + 4.76T + 11T^{2} \)
13 \( 1 + 3.91T + 13T^{2} \)
17 \( 1 - 3.31T + 17T^{2} \)
19 \( 1 + 1.85T + 19T^{2} \)
23 \( 1 + 1.54T + 23T^{2} \)
29 \( 1 - 8.87T + 29T^{2} \)
31 \( 1 - 9.75T + 31T^{2} \)
41 \( 1 + 5.06T + 41T^{2} \)
43 \( 1 + 9.99T + 43T^{2} \)
47 \( 1 - 4.82T + 47T^{2} \)
53 \( 1 + 5.13T + 53T^{2} \)
59 \( 1 + 1.05T + 59T^{2} \)
61 \( 1 + 4.14T + 61T^{2} \)
67 \( 1 - 1.00T + 67T^{2} \)
71 \( 1 - 6.45T + 71T^{2} \)
73 \( 1 - 10.7T + 73T^{2} \)
79 \( 1 + 1.19T + 79T^{2} \)
83 \( 1 - 10.6T + 83T^{2} \)
89 \( 1 - 7.29T + 89T^{2} \)
97 \( 1 - 14.8T + 97T^{2} \)
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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

−9.798829738217748380051029947283, −8.237634164788602682644172560848, −7.31485052323480244636394400067, −6.44382716403366498822880228439, −6.15252433879514551543924589270, −5.08319069364068668473291965189, −4.75367330997839420357848093515, −3.35127473752737026006467906167, −2.50959940832418229792012443377, −0.55536893386650606231007681147, 0.55536893386650606231007681147, 2.50959940832418229792012443377, 3.35127473752737026006467906167, 4.75367330997839420357848093515, 5.08319069364068668473291965189, 6.15252433879514551543924589270, 6.44382716403366498822880228439, 7.31485052323480244636394400067, 8.237634164788602682644172560848, 9.798829738217748380051029947283

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