Properties

Label 2-2200-1.1-c3-0-118
Degree $2$
Conductor $2200$
Sign $-1$
Analytic cond. $129.804$
Root an. cond. $11.3931$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·3-s − 8·7-s − 11·9-s + 11·11-s + 58·13-s − 114·17-s − 4·19-s − 32·21-s + 152·23-s − 152·27-s − 138·29-s + 208·31-s + 44·33-s + 226·37-s + 232·39-s − 294·41-s − 276·43-s + 240·47-s − 279·49-s − 456·51-s + 370·53-s − 16·57-s − 716·59-s − 650·61-s + 88·63-s − 124·67-s + 608·69-s + ⋯
L(s)  = 1  + 0.769·3-s − 0.431·7-s − 0.407·9-s + 0.301·11-s + 1.23·13-s − 1.62·17-s − 0.0482·19-s − 0.332·21-s + 1.37·23-s − 1.08·27-s − 0.883·29-s + 1.20·31-s + 0.232·33-s + 1.00·37-s + 0.952·39-s − 1.11·41-s − 0.978·43-s + 0.744·47-s − 0.813·49-s − 1.25·51-s + 0.958·53-s − 0.0371·57-s − 1.57·59-s − 1.36·61-s + 0.175·63-s − 0.226·67-s + 1.06·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2200 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2200\)    =    \(2^{3} \cdot 5^{2} \cdot 11\)
Sign: $-1$
Analytic conductor: \(129.804\)
Root analytic conductor: \(11.3931\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 2200,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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 \( 1 \)
11 \( 1 - p T \)
good3 \( 1 - 4 T + p^{3} T^{2} \)
7 \( 1 + 8 T + p^{3} T^{2} \)
13 \( 1 - 58 T + p^{3} T^{2} \)
17 \( 1 + 114 T + p^{3} T^{2} \)
19 \( 1 + 4 T + p^{3} T^{2} \)
23 \( 1 - 152 T + p^{3} T^{2} \)
29 \( 1 + 138 T + p^{3} T^{2} \)
31 \( 1 - 208 T + p^{3} T^{2} \)
37 \( 1 - 226 T + p^{3} T^{2} \)
41 \( 1 + 294 T + p^{3} T^{2} \)
43 \( 1 + 276 T + p^{3} T^{2} \)
47 \( 1 - 240 T + p^{3} T^{2} \)
53 \( 1 - 370 T + p^{3} T^{2} \)
59 \( 1 + 716 T + p^{3} T^{2} \)
61 \( 1 + 650 T + p^{3} T^{2} \)
67 \( 1 + 124 T + p^{3} T^{2} \)
71 \( 1 - 232 T + p^{3} T^{2} \)
73 \( 1 - 454 T + p^{3} T^{2} \)
79 \( 1 + 144 T + p^{3} T^{2} \)
83 \( 1 - 692 T + p^{3} T^{2} \)
89 \( 1 + 1206 T + p^{3} T^{2} \)
97 \( 1 - 1438 T + p^{3} T^{2} \)
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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

−8.527529157093091592336940100729, −7.67846125709358158555464627529, −6.62208679427469008721777810867, −6.20632124128393342836401490447, −5.04822530294021517508229219619, −4.06291029144313208314643976342, −3.26668494206476823210081647916, −2.49352628878785471837518366657, −1.35832727465456428928216258711, 0, 1.35832727465456428928216258711, 2.49352628878785471837518366657, 3.26668494206476823210081647916, 4.06291029144313208314643976342, 5.04822530294021517508229219619, 6.20632124128393342836401490447, 6.62208679427469008721777810867, 7.67846125709358158555464627529, 8.527529157093091592336940100729

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