Properties

Label 2-1848-1.1-c1-0-30
Degree $2$
Conductor $1848$
Sign $-1$
Analytic cond. $14.7563$
Root an. cond. $3.84140$
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  + 3-s − 5-s + 7-s + 9-s − 11-s − 5·13-s − 15-s − 6·17-s − 19-s + 21-s + 4·23-s − 4·25-s + 27-s + 29-s − 10·31-s − 33-s − 35-s + 37-s − 5·39-s − 8·43-s − 45-s + 47-s + 49-s − 6·51-s + 8·53-s + 55-s − 57-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s + 0.377·7-s + 1/3·9-s − 0.301·11-s − 1.38·13-s − 0.258·15-s − 1.45·17-s − 0.229·19-s + 0.218·21-s + 0.834·23-s − 4/5·25-s + 0.192·27-s + 0.185·29-s − 1.79·31-s − 0.174·33-s − 0.169·35-s + 0.164·37-s − 0.800·39-s − 1.21·43-s − 0.149·45-s + 0.145·47-s + 1/7·49-s − 0.840·51-s + 1.09·53-s + 0.134·55-s − 0.132·57-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1848 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1848 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1848\)    =    \(2^{3} \cdot 3 \cdot 7 \cdot 11\)
Sign: $-1$
Analytic conductor: \(14.7563\)
Root analytic conductor: \(3.84140\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1848,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
7 \( 1 - T \)
11 \( 1 + T \)
good5 \( 1 + T + p T^{2} \)
13 \( 1 + 5 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - T + p T^{2} \)
31 \( 1 + 10 T + p T^{2} \)
37 \( 1 - T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 - T + p T^{2} \)
53 \( 1 - 8 T + p T^{2} \)
59 \( 1 + 3 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 - 13 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 3 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 2 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 16 T + p T^{2} \)
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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

−8.849850990040599939674459244918, −8.061330974970410988557609129102, −7.32906257514559404736559055822, −6.76467133439150651158112755625, −5.43842743581863918516389791646, −4.66431671874617585722979564988, −3.86531234360204512054976650332, −2.71194579855237990031082889907, −1.88795086058624889483632780490, 0, 1.88795086058624889483632780490, 2.71194579855237990031082889907, 3.86531234360204512054976650332, 4.66431671874617585722979564988, 5.43842743581863918516389791646, 6.76467133439150651158112755625, 7.32906257514559404736559055822, 8.061330974970410988557609129102, 8.849850990040599939674459244918

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