Properties

Label 2-1850-1.1-c1-0-17
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 − 1.44·3-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: $\chi_{1850} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1850,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.339899330\)
\(L(\frac12)\) \(\approx\) \(2.339899330\)
\(L(\frac{3}{2})\) 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

−9.185075631281304424842418069504, −8.396879899200597943631683923191, −7.53204207622829071181453656909, −6.56364957581995654575158709468, −6.00293497616628195120104017780, −5.13932141230260486730540470487, −4.50299375581603978348995533716, −3.54110406293956700135370985156, −2.26511061878540504975212372415, −1.03133138935126130223078502192, 1.03133138935126130223078502192, 2.26511061878540504975212372415, 3.54110406293956700135370985156, 4.50299375581603978348995533716, 5.13932141230260486730540470487, 6.00293497616628195120104017780, 6.56364957581995654575158709468, 7.53204207622829071181453656909, 8.396879899200597943631683923191, 9.185075631281304424842418069504

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