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Properties

Label 2-342720-1.1-c1-0-317
Degree $2$
Conductor $342720$
Sign $-1$
Analytic cond. $2736.63$
Root an. cond. $52.3128$
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 − 7-s − 2·11-s + 5·13-s − 17-s + 2·19-s − 23-s + 25-s + 8·29-s − 31-s + 35-s + 3·37-s + 7·41-s − 47-s + 49-s − 8·53-s + 2·55-s + 7·61-s − 5·65-s + 16·67-s − 10·71-s − 8·73-s + 2·77-s + 6·79-s + 13·83-s + 85-s + 2·89-s + ⋯
L(s)  = 1  − 0.447·5-s − 0.377·7-s − 0.603·11-s + 1.38·13-s − 0.242·17-s + 0.458·19-s − 0.208·23-s + 1/5·25-s + 1.48·29-s − 0.179·31-s + 0.169·35-s + 0.493·37-s + 1.09·41-s − 0.145·47-s + 1/7·49-s − 1.09·53-s + 0.269·55-s + 0.896·61-s − 0.620·65-s + 1.95·67-s − 1.18·71-s − 0.936·73-s + 0.227·77-s + 0.675·79-s + 1.42·83-s + 0.108·85-s + 0.211·89-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 342720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 342720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(342720\)    =    \(2^{6} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 17\)
Sign: $-1$
Analytic conductor: \(2736.63\)
Root analytic conductor: \(52.3128\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 342720,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
17 \( 1 + T \)
good11 \( 1 + 2 T + p T^{2} \)
13 \( 1 - 5 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + T + p T^{2} \)
29 \( 1 - 8 T + p T^{2} \)
31 \( 1 + T + p T^{2} \)
37 \( 1 - 3 T + p T^{2} \)
41 \( 1 - 7 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 + T + p T^{2} \)
53 \( 1 + 8 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 7 T + p T^{2} \)
67 \( 1 - 16 T + p T^{2} \)
71 \( 1 + 10 T + p T^{2} \)
73 \( 1 + 8 T + p T^{2} \)
79 \( 1 - 6 T + p T^{2} \)
83 \( 1 - 13 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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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

−12.82049476859812, −12.33555575075725, −11.94965647115076, −11.24450165105312, −11.09871253535600, −10.62577062127783, −10.02362281805844, −9.703240259413272, −9.084696833016122, −8.594142786230126, −8.271699141151763, −7.754774016407130, −7.335963716212150, −6.684398086950570, −6.267918920171758, −5.922500168234977, −5.235655997583022, −4.780825274434374, −4.192487475725436, −3.701278474610572, −3.232384907389159, −2.674269495109300, −2.136661681060869, −1.221575403399886, −0.8313266592609209, 0, 0.8313266592609209, 1.221575403399886, 2.136661681060869, 2.674269495109300, 3.232384907389159, 3.701278474610572, 4.192487475725436, 4.780825274434374, 5.235655997583022, 5.922500168234977, 6.267918920171758, 6.684398086950570, 7.335963716212150, 7.754774016407130, 8.271699141151763, 8.594142786230126, 9.084696833016122, 9.703240259413272, 10.02362281805844, 10.62577062127783, 11.09871253535600, 11.24450165105312, 11.94965647115076, 12.33555575075725, 12.82049476859812

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