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Properties

Label 2-30960-1.1-c1-0-29
Degree $2$
Conductor $30960$
Sign $-1$
Analytic cond. $247.216$
Root an. cond. $15.7231$
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 − 4·7-s + 4·11-s + 4·13-s − 4·23-s + 25-s + 6·29-s + 4·31-s + 4·35-s − 2·37-s − 2·41-s − 43-s − 8·47-s + 9·49-s + 6·53-s − 4·55-s − 4·59-s − 4·61-s − 4·65-s − 8·67-s + 4·71-s − 10·73-s − 16·77-s + 8·79-s − 14·83-s − 12·89-s − 16·91-s + ⋯
L(s)  = 1  − 0.447·5-s − 1.51·7-s + 1.20·11-s + 1.10·13-s − 0.834·23-s + 1/5·25-s + 1.11·29-s + 0.718·31-s + 0.676·35-s − 0.328·37-s − 0.312·41-s − 0.152·43-s − 1.16·47-s + 9/7·49-s + 0.824·53-s − 0.539·55-s − 0.520·59-s − 0.512·61-s − 0.496·65-s − 0.977·67-s + 0.474·71-s − 1.17·73-s − 1.82·77-s + 0.900·79-s − 1.53·83-s − 1.27·89-s − 1.67·91-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 30960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(30960\)    =    \(2^{4} \cdot 3^{2} \cdot 5 \cdot 43\)
Sign: $-1$
Analytic conductor: \(247.216\)
Root analytic conductor: \(15.7231\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 30960,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
43 \( 1 + T \)
good7 \( 1 + 4 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 4 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 14 T + p T^{2} \)
89 \( 1 + 12 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−15.35451460516648, −15.01951209292682, −14.11561130254802, −13.82291075617352, −13.27381366886866, −12.69210478530790, −12.10227667017102, −11.80392237957803, −11.15913149392145, −10.43880689955508, −9.985910472050522, −9.462923879241432, −8.817927053652736, −8.441984975451130, −7.744183097673437, −6.844741064674756, −6.570601255320627, −6.123002185505497, −5.448510553981631, −4.328733141470178, −4.075468095991014, −3.242217666701449, −2.960394605120150, −1.744005664925133, −0.9642575368839718, 0, 0.9642575368839718, 1.744005664925133, 2.960394605120150, 3.242217666701449, 4.075468095991014, 4.328733141470178, 5.448510553981631, 6.123002185505497, 6.570601255320627, 6.844741064674756, 7.744183097673437, 8.441984975451130, 8.817927053652736, 9.462923879241432, 9.985910472050522, 10.43880689955508, 11.15913149392145, 11.80392237957803, 12.10227667017102, 12.69210478530790, 13.27381366886866, 13.82291075617352, 14.11561130254802, 15.01951209292682, 15.35451460516648

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