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Properties

Label 2-1900-19.18-c0-0-0
Degree $2$
Conductor $1900$
Sign $1$
Analytic cond. $0.948223$
Root an. cond. $0.973767$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.73·7-s + 9-s + 11-s + 1.73·17-s − 19-s + 1.73·43-s + 1.73·47-s + 1.99·49-s + 61-s − 1.73·63-s − 1.73·73-s − 1.73·77-s + 81-s + 99-s − 2·101-s − 2.99·119-s + ⋯
L(s)  = 1  − 1.73·7-s + 9-s + 11-s + 1.73·17-s − 19-s + 1.73·43-s + 1.73·47-s + 1.99·49-s + 61-s − 1.73·63-s − 1.73·73-s − 1.73·77-s + 81-s + 99-s − 2·101-s − 2.99·119-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1900\)    =    \(2^{2} \cdot 5^{2} \cdot 19\)
Sign: $1$
Analytic conductor: \(0.948223\)
Root analytic conductor: \(0.973767\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1900} (1101, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1900,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.109946567\)
\(L(\frac12)\) \(\approx\) \(1.109946567\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
19 \( 1 + T \)
good3 \( 1 - T^{2} \)
7 \( 1 + 1.73T + T^{2} \)
11 \( 1 - T + T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 - 1.73T + T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - 1.73T + T^{2} \)
47 \( 1 - 1.73T + T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - T + T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + 1.73T + T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 - T^{2} \)
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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

−9.450591814646781873203051794884, −8.870064979481027082270643092220, −7.65684540619989280652809414657, −6.98630077194399233747880722870, −6.29282720143167857946795516937, −5.60417393405923755434411622934, −4.17308416120135586916665081053, −3.69563145749777309301725343961, −2.62565976578020905136690172164, −1.12614990228210544562975764237, 1.12614990228210544562975764237, 2.62565976578020905136690172164, 3.69563145749777309301725343961, 4.17308416120135586916665081053, 5.60417393405923755434411622934, 6.29282720143167857946795516937, 6.98630077194399233747880722870, 7.65684540619989280652809414657, 8.870064979481027082270643092220, 9.450591814646781873203051794884

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