Properties

Label 2-1900-1.1-c1-0-14
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3.28·3-s − 1.93·7-s + 7.80·9-s + 5.62·11-s − 2.07·13-s + 3.42·17-s + 19-s − 6.35·21-s − 5.35·23-s + 15.7·27-s + 1.09·29-s + 3.09·31-s + 18.5·33-s + 3.28·37-s − 6.80·39-s − 11.6·41-s − 0.501·43-s − 12.6·47-s − 3.26·49-s + 11.2·51-s + 3.01·53-s + 3.28·57-s − 2.35·59-s + 7.62·61-s − 15.0·63-s + 12.2·67-s − 17.6·69-s + ⋯
L(s)  = 1  + 1.89·3-s − 0.730·7-s + 2.60·9-s + 1.69·11-s − 0.574·13-s + 0.830·17-s + 0.229·19-s − 1.38·21-s − 1.11·23-s + 3.03·27-s + 0.202·29-s + 0.555·31-s + 3.22·33-s + 0.540·37-s − 1.08·39-s − 1.81·41-s − 0.0764·43-s − 1.84·47-s − 0.466·49-s + 1.57·51-s + 0.413·53-s + 0.435·57-s − 0.306·59-s + 0.976·61-s − 1.89·63-s + 1.49·67-s − 2.12·69-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: \(0\)
Selberg data: \((2,\ 1900,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.617005029\)
\(L(\frac12)\) \(\approx\) \(3.617005029\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.28T + 3T^{2} \)
7 \( 1 + 1.93T + 7T^{2} \)
11 \( 1 - 5.62T + 11T^{2} \)
13 \( 1 + 2.07T + 13T^{2} \)
17 \( 1 - 3.42T + 17T^{2} \)
23 \( 1 + 5.35T + 23T^{2} \)
29 \( 1 - 1.09T + 29T^{2} \)
31 \( 1 - 3.09T + 31T^{2} \)
37 \( 1 - 3.28T + 37T^{2} \)
41 \( 1 + 11.6T + 41T^{2} \)
43 \( 1 + 0.501T + 43T^{2} \)
47 \( 1 + 12.6T + 47T^{2} \)
53 \( 1 - 3.01T + 53T^{2} \)
59 \( 1 + 2.35T + 59T^{2} \)
61 \( 1 - 7.62T + 61T^{2} \)
67 \( 1 - 12.2T + 67T^{2} \)
71 \( 1 - 14.3T + 71T^{2} \)
73 \( 1 + 2.87T + 73T^{2} \)
79 \( 1 - 4T + 79T^{2} \)
83 \( 1 - 5.35T + 83T^{2} \)
89 \( 1 - 8.35T + 89T^{2} \)
97 \( 1 - 3.01T + 97T^{2} \)
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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

−9.333289615233141188616662535514, −8.373860710128365146512028189701, −7.924619990615411977971366935903, −6.87701303704460933934273181062, −6.43933901159237273433225510714, −4.89441625447526179531833013217, −3.76938485448122477379246609713, −3.46653409816443336706769462798, −2.37422277111233868413714740535, −1.35757910354703740290716756116, 1.35757910354703740290716756116, 2.37422277111233868413714740535, 3.46653409816443336706769462798, 3.76938485448122477379246609713, 4.89441625447526179531833013217, 6.43933901159237273433225510714, 6.87701303704460933934273181062, 7.924619990615411977971366935903, 8.373860710128365146512028189701, 9.333289615233141188616662535514

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