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Properties

Label 2-2268-1.1-c1-0-18
Degree $2$
Conductor $2268$
Sign $-1$
Analytic cond. $18.1100$
Root an. cond. $4.25559$
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  + 7-s − 3·11-s − 4·13-s + 6·17-s − 4·19-s − 5·25-s + 6·29-s + 2·31-s − 7·37-s − 12·41-s − 7·43-s − 6·47-s + 49-s + 3·53-s − 6·59-s + 2·61-s − 13·67-s + 9·71-s + 8·73-s − 3·77-s − 79-s + 12·89-s − 4·91-s − 4·97-s − 12·101-s − 4·103-s + 15·107-s + ⋯
L(s)  = 1  + 0.377·7-s − 0.904·11-s − 1.10·13-s + 1.45·17-s − 0.917·19-s − 25-s + 1.11·29-s + 0.359·31-s − 1.15·37-s − 1.87·41-s − 1.06·43-s − 0.875·47-s + 1/7·49-s + 0.412·53-s − 0.781·59-s + 0.256·61-s − 1.58·67-s + 1.06·71-s + 0.936·73-s − 0.341·77-s − 0.112·79-s + 1.27·89-s − 0.419·91-s − 0.406·97-s − 1.19·101-s − 0.394·103-s + 1.45·107-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2268\)    =    \(2^{2} \cdot 3^{4} \cdot 7\)
Sign: $-1$
Analytic conductor: \(18.1100\)
Root analytic conductor: \(4.25559\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 2268,\ (\ :1/2),\ -1)\)

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 \( 1 \)
7 \( 1 - T \)
good5 \( 1 + p T^{2} \)
11 \( 1 + 3 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 + 7 T + p T^{2} \)
41 \( 1 + 12 T + p T^{2} \)
43 \( 1 + 7 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 - 3 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 13 T + p T^{2} \)
71 \( 1 - 9 T + p T^{2} \)
73 \( 1 - 8 T + p T^{2} \)
79 \( 1 + T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 12 T + p T^{2} \)
97 \( 1 + 4 T + p T^{2} \)
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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

−8.367405229597541969143450600182, −8.013715186180625594119400228016, −7.16969185229083907061004830786, −6.31891530104611065760601551687, −5.23813287136191298515733096075, −4.87092313996227080026581484887, −3.64041872315825866632821982347, −2.69339082526379787317114374377, −1.66109110854626091794561079002, 0, 1.66109110854626091794561079002, 2.69339082526379787317114374377, 3.64041872315825866632821982347, 4.87092313996227080026581484887, 5.23813287136191298515733096075, 6.31891530104611065760601551687, 7.16969185229083907061004830786, 8.013715186180625594119400228016, 8.367405229597541969143450600182

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