Properties

Label 2-1872-1.1-c1-0-25
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·5-s − 4·7-s + 13-s − 2·17-s − 8·19-s + 8·23-s − 25-s + 2·29-s − 4·31-s − 8·35-s − 10·37-s − 2·41-s + 4·43-s − 12·47-s + 9·49-s − 6·53-s − 2·61-s + 2·65-s − 8·67-s − 12·71-s + 10·73-s + 8·79-s − 4·85-s + 14·89-s − 4·91-s − 16·95-s + 2·97-s + ⋯
L(s)  = 1  + 0.894·5-s − 1.51·7-s + 0.277·13-s − 0.485·17-s − 1.83·19-s + 1.66·23-s − 1/5·25-s + 0.371·29-s − 0.718·31-s − 1.35·35-s − 1.64·37-s − 0.312·41-s + 0.609·43-s − 1.75·47-s + 9/7·49-s − 0.824·53-s − 0.256·61-s + 0.248·65-s − 0.977·67-s − 1.42·71-s + 1.17·73-s + 0.900·79-s − 0.433·85-s + 1.48·89-s − 0.419·91-s − 1.64·95-s + 0.203·97-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: \(1\)
Selberg data: \((2,\ 1872,\ (\ :1/2),\ -1)\)

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 \( 1 \)
3 \( 1 \)
13 \( 1 - T \)
good5 \( 1 - 2 T + p T^{2} \)
7 \( 1 + 4 T + p T^{2} \)
11 \( 1 + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
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 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−9.158792946221366931900105912370, −8.209770136822864596347152768350, −6.84717778862769781936733328882, −6.57432395350568485912302103534, −5.78674978184567296447754787702, −4.80441249928361298007080575834, −3.67113970451719789938663633179, −2.81347328384033747404310399701, −1.75350296463647814243579299746, 0, 1.75350296463647814243579299746, 2.81347328384033747404310399701, 3.67113970451719789938663633179, 4.80441249928361298007080575834, 5.78674978184567296447754787702, 6.57432395350568485912302103534, 6.84717778862769781936733328882, 8.209770136822864596347152768350, 9.158792946221366931900105912370

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