Properties

Label 2-1872-1.1-c1-0-9
Degree $2$
Conductor $1872$
Sign $1$
Analytic cond. $14.9479$
Root an. cond. $3.86626$
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  − 2·5-s + 4·7-s + 4·11-s + 13-s − 2·17-s − 25-s + 10·29-s − 4·31-s − 8·35-s − 2·37-s − 6·41-s + 12·43-s + 9·49-s − 6·53-s − 8·55-s + 12·59-s − 2·61-s − 2·65-s + 8·67-s + 2·73-s + 16·77-s − 8·79-s + 4·83-s + 4·85-s + 2·89-s + 4·91-s + 10·97-s + ⋯
L(s)  = 1  − 0.894·5-s + 1.51·7-s + 1.20·11-s + 0.277·13-s − 0.485·17-s − 1/5·25-s + 1.85·29-s − 0.718·31-s − 1.35·35-s − 0.328·37-s − 0.937·41-s + 1.82·43-s + 9/7·49-s − 0.824·53-s − 1.07·55-s + 1.56·59-s − 0.256·61-s − 0.248·65-s + 0.977·67-s + 0.234·73-s + 1.82·77-s − 0.900·79-s + 0.439·83-s + 0.433·85-s + 0.211·89-s + 0.419·91-s + 1.01·97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1872\)    =    \(2^{4} \cdot 3^{2} \cdot 13\)
Sign: $1$
Analytic conductor: \(14.9479\)
Root analytic conductor: \(3.86626\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1872,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.909153812\)
\(L(\frac12)\) \(\approx\) \(1.909153812\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
13 \( 1 - T \)
good5 \( 1 + 2 T + p T^{2} \)
7 \( 1 - 4 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 10 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−8.921039436304358900231528874001, −8.484648143275568894036347393857, −7.73604447848719394517643815023, −6.99405970659997290412829823598, −6.08826329831047007406167672262, −4.94568196736683705255358366288, −4.31391709618999720844014512009, −3.55158930907206838697007657967, −2.10208151844778724001409156325, −1.00171028893857761930540835980, 1.00171028893857761930540835980, 2.10208151844778724001409156325, 3.55158930907206838697007657967, 4.31391709618999720844014512009, 4.94568196736683705255358366288, 6.08826329831047007406167672262, 6.99405970659997290412829823598, 7.73604447848719394517643815023, 8.484648143275568894036347393857, 8.921039436304358900231528874001

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