Properties

Label 2-1850-1.1-c1-0-20
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 − 0.732·3-s + 4-s − 0.732·6-s + 4.73·7-s + 8-s − 2.46·9-s − 5.46·11-s − 0.732·12-s + 5.46·13-s + 4.73·14-s + 16-s − 5.46·17-s − 2.46·18-s + 6.19·19-s − 3.46·21-s − 5.46·22-s + 8·23-s − 0.732·24-s + 5.46·26-s + 4·27-s + 4.73·28-s + 4.92·29-s + 0.732·31-s + 32-s + 4·33-s − 5.46·34-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.422·3-s + 0.5·4-s − 0.298·6-s + 1.78·7-s + 0.353·8-s − 0.821·9-s − 1.64·11-s − 0.211·12-s + 1.51·13-s + 1.26·14-s + 0.250·16-s − 1.32·17-s − 0.580·18-s + 1.42·19-s − 0.755·21-s − 1.16·22-s + 1.66·23-s − 0.149·24-s + 1.07·26-s + 0.769·27-s + 0.894·28-s + 0.915·29-s + 0.131·31-s + 0.176·32-s + 0.696·33-s − 0.937·34-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

\(L(1)\) \(\approx\) \(2.739274093\)
\(L(\frac12)\) \(\approx\) \(2.739274093\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.732T + 3T^{2} \)
7 \( 1 - 4.73T + 7T^{2} \)
11 \( 1 + 5.46T + 11T^{2} \)
13 \( 1 - 5.46T + 13T^{2} \)
17 \( 1 + 5.46T + 17T^{2} \)
19 \( 1 - 6.19T + 19T^{2} \)
23 \( 1 - 8T + 23T^{2} \)
29 \( 1 - 4.92T + 29T^{2} \)
31 \( 1 - 0.732T + 31T^{2} \)
41 \( 1 + 2T + 41T^{2} \)
43 \( 1 + 6.92T + 43T^{2} \)
47 \( 1 - 4.73T + 47T^{2} \)
53 \( 1 - 6T + 53T^{2} \)
59 \( 1 + 10.1T + 59T^{2} \)
61 \( 1 + 4.92T + 61T^{2} \)
67 \( 1 - 3.66T + 67T^{2} \)
71 \( 1 - 2.92T + 71T^{2} \)
73 \( 1 - 0.928T + 73T^{2} \)
79 \( 1 - 8.73T + 79T^{2} \)
83 \( 1 - 8.73T + 83T^{2} \)
89 \( 1 + 2T + 89T^{2} \)
97 \( 1 - 2T + 97T^{2} \)
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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

−8.963076370905079983796561663384, −8.357825002388308440132058755782, −7.72120075724208239820633257686, −6.75436080125806691900573851388, −5.73891220942164057182818500067, −5.08025389328291495328963856726, −4.70847845368273107965005237560, −3.31006486743633003984378411175, −2.39039497173098737359083859503, −1.10706962114590556341978402535, 1.10706962114590556341978402535, 2.39039497173098737359083859503, 3.31006486743633003984378411175, 4.70847845368273107965005237560, 5.08025389328291495328963856726, 5.73891220942164057182818500067, 6.75436080125806691900573851388, 7.72120075724208239820633257686, 8.357825002388308440132058755782, 8.963076370905079983796561663384

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