Properties

Label 2-1900-1.1-c1-0-26
Degree $2$
Conductor $1900$
Sign $-1$
Analytic cond. $15.1715$
Root an. cond. $3.89507$
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  + 0.874·3-s + 2.82·7-s − 2.23·9-s − 0.763·11-s − 5.45·13-s − 7.40·17-s − 19-s + 2.47·21-s − 1.08·23-s − 4.57·27-s − 4.47·29-s − 4·31-s − 0.667·33-s − 2.62·37-s − 4.76·39-s − 6·41-s + 8.48·43-s + 8.48·47-s + 1.00·49-s − 6.47·51-s + 2.62·53-s − 0.874·57-s − 1.52·59-s − 11.7·61-s − 6.32·63-s + 11.1·67-s − 0.944·69-s + ⋯
L(s)  = 1  + 0.504·3-s + 1.06·7-s − 0.745·9-s − 0.230·11-s − 1.51·13-s − 1.79·17-s − 0.229·19-s + 0.539·21-s − 0.225·23-s − 0.880·27-s − 0.830·29-s − 0.718·31-s − 0.116·33-s − 0.431·37-s − 0.762·39-s − 0.937·41-s + 1.29·43-s + 1.23·47-s + 0.142·49-s − 0.906·51-s + 0.360·53-s − 0.115·57-s − 0.198·59-s − 1.49·61-s − 0.796·63-s + 1.35·67-s − 0.113·69-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1900\)    =    \(2^{2} \cdot 5^{2} \cdot 19\)
Sign: $-1$
Analytic conductor: \(15.1715\)
Root analytic conductor: \(3.89507\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1900} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1900,\ (\ :1/2),\ -1)\)

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 \( 1 \)
5 \( 1 \)
19 \( 1 + T \)
good3 \( 1 - 0.874T + 3T^{2} \)
7 \( 1 - 2.82T + 7T^{2} \)
11 \( 1 + 0.763T + 11T^{2} \)
13 \( 1 + 5.45T + 13T^{2} \)
17 \( 1 + 7.40T + 17T^{2} \)
23 \( 1 + 1.08T + 23T^{2} \)
29 \( 1 + 4.47T + 29T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 + 2.62T + 37T^{2} \)
41 \( 1 + 6T + 41T^{2} \)
43 \( 1 - 8.48T + 43T^{2} \)
47 \( 1 - 8.48T + 47T^{2} \)
53 \( 1 - 2.62T + 53T^{2} \)
59 \( 1 + 1.52T + 59T^{2} \)
61 \( 1 + 11.7T + 61T^{2} \)
67 \( 1 - 11.1T + 67T^{2} \)
71 \( 1 + 10.4T + 71T^{2} \)
73 \( 1 - 5.24T + 73T^{2} \)
79 \( 1 - 15.4T + 79T^{2} \)
83 \( 1 - 13.7T + 83T^{2} \)
89 \( 1 - 2.94T + 89T^{2} \)
97 \( 1 - 13.9T + 97T^{2} \)
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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

−8.976716425638819219527418182138, −7.962217379475188451136099364788, −7.50528343732638310397045379712, −6.52365139204136561985983629483, −5.40647311620450029780470179093, −4.78452359403743836035818973320, −3.84981833633243038529077997270, −2.51339429226187543878896182769, −2.01505303799945403227893738935, 0, 2.01505303799945403227893738935, 2.51339429226187543878896182769, 3.84981833633243038529077997270, 4.78452359403743836035818973320, 5.40647311620450029780470179093, 6.52365139204136561985983629483, 7.50528343732638310397045379712, 7.962217379475188451136099364788, 8.976716425638819219527418182138

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