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Properties

Label 2-141120-1.1-c1-0-131
Degree $2$
Conductor $141120$
Sign $-1$
Analytic cond. $1126.84$
Root an. cond. $33.5685$
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  − 5-s − 6·11-s − 4·13-s − 6·17-s + 4·19-s + 25-s − 6·29-s − 4·31-s − 8·37-s + 8·43-s − 6·53-s + 6·55-s + 6·59-s + 2·61-s + 4·65-s − 4·67-s − 12·71-s + 10·73-s + 4·79-s + 12·83-s + 6·85-s + 12·89-s − 4·95-s − 2·97-s + 101-s + 103-s + 107-s + ⋯
L(s)  = 1  − 0.447·5-s − 1.80·11-s − 1.10·13-s − 1.45·17-s + 0.917·19-s + 1/5·25-s − 1.11·29-s − 0.718·31-s − 1.31·37-s + 1.21·43-s − 0.824·53-s + 0.809·55-s + 0.781·59-s + 0.256·61-s + 0.496·65-s − 0.488·67-s − 1.42·71-s + 1.17·73-s + 0.450·79-s + 1.31·83-s + 0.650·85-s + 1.27·89-s − 0.410·95-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 141120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 141120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(141120\)    =    \(2^{6} \cdot 3^{2} \cdot 5 \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(1126.84\)
Root analytic conductor: \(33.5685\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 141120,\ (\ :1/2),\ -1)\)

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 \( 1 \)
5 \( 1 + T \)
7 \( 1 \)
good11 \( 1 + 6 T + p T^{2} \)
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 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 12 T + p T^{2} \)
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

−13.52045754373864, −13.06223611523743, −12.84752370425266, −12.15438512382024, −11.82004482346627, −11.14525102166992, −10.74373604327758, −10.44659592674543, −9.773876062152291, −9.237457081872697, −8.902842016934432, −8.135960566931235, −7.729826644484478, −7.339635815438120, −6.954479884965123, −6.215523707099944, −5.486480874030236, −5.133681306375647, −4.749190936523532, −4.022539241710126, −3.432198702830807, −2.753037531221799, −2.304863143969626, −1.739593060095290, −0.5448183661011304, 0, 0.5448183661011304, 1.739593060095290, 2.304863143969626, 2.753037531221799, 3.432198702830807, 4.022539241710126, 4.749190936523532, 5.133681306375647, 5.486480874030236, 6.215523707099944, 6.954479884965123, 7.339635815438120, 7.729826644484478, 8.135960566931235, 8.902842016934432, 9.237457081872697, 9.773876062152291, 10.44659592674543, 10.74373604327758, 11.14525102166992, 11.82004482346627, 12.15438512382024, 12.84752370425266, 13.06223611523743, 13.52045754373864

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