Properties

Label 2-1900-1.1-c1-0-6
Degree $2$
Conductor $1900$
Sign $1$
Analytic cond. $15.1715$
Root an. cond. $3.89507$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s + 3·7-s + 9-s + 5·11-s + 4·13-s + 3·17-s − 19-s − 6·21-s − 8·23-s + 4·27-s − 2·29-s + 4·31-s − 10·33-s − 10·37-s − 8·39-s + 10·41-s − 43-s + 47-s + 2·49-s − 6·51-s + 4·53-s + 2·57-s + 6·59-s − 13·61-s + 3·63-s + 12·67-s + 16·69-s + ⋯
L(s)  = 1  − 1.15·3-s + 1.13·7-s + 1/3·9-s + 1.50·11-s + 1.10·13-s + 0.727·17-s − 0.229·19-s − 1.30·21-s − 1.66·23-s + 0.769·27-s − 0.371·29-s + 0.718·31-s − 1.74·33-s − 1.64·37-s − 1.28·39-s + 1.56·41-s − 0.152·43-s + 0.145·47-s + 2/7·49-s − 0.840·51-s + 0.549·53-s + 0.264·57-s + 0.781·59-s − 1.66·61-s + 0.377·63-s + 1.46·67-s + 1.92·69-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: yes
Arithmetic: yes
Character: $\chi_{1900} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1900,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.489784422\)
\(L(\frac12)\) \(\approx\) \(1.489784422\)
\(L(\frac{3}{2})\) 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 \( 1 + 2 T + p T^{2} \)
7 \( 1 - 3 T + p T^{2} \)
11 \( 1 - 5 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + T + p T^{2} \)
47 \( 1 - T + p T^{2} \)
53 \( 1 - 4 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 + 13 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 2 T + p T^{2} \)
73 \( 1 + 9 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 12 T + p T^{2} \)
97 \( 1 - 8 T + p T^{2} \)
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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

−9.142669581834378698595686186454, −8.426347554243774025274773727164, −7.67036876906477496457872273315, −6.54496232916405090183568591347, −6.05356181637470555300777284468, −5.29796420868215321731869729709, −4.34576334774572475342173538720, −3.62036505864368699268002729080, −1.86326488698354633791416853460, −0.942787934452988901616816885952, 0.942787934452988901616816885952, 1.86326488698354633791416853460, 3.62036505864368699268002729080, 4.34576334774572475342173538720, 5.29796420868215321731869729709, 6.05356181637470555300777284468, 6.54496232916405090183568591347, 7.67036876906477496457872273315, 8.426347554243774025274773727164, 9.142669581834378698595686186454

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