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Properties

Label 2-342720-1.1-c1-0-122
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5-s − 7-s + 6·11-s − 17-s − 6·19-s + 4·23-s + 25-s − 4·29-s + 8·31-s − 35-s − 4·37-s + 2·41-s + 12·43-s − 8·47-s + 49-s − 6·53-s + 6·55-s + 14·59-s − 2·61-s − 8·67-s − 6·73-s − 6·77-s − 4·79-s + 6·83-s − 85-s − 6·89-s − 6·95-s + ⋯
L(s)  = 1  + 0.447·5-s − 0.377·7-s + 1.80·11-s − 0.242·17-s − 1.37·19-s + 0.834·23-s + 1/5·25-s − 0.742·29-s + 1.43·31-s − 0.169·35-s − 0.657·37-s + 0.312·41-s + 1.82·43-s − 1.16·47-s + 1/7·49-s − 0.824·53-s + 0.809·55-s + 1.82·59-s − 0.256·61-s − 0.977·67-s − 0.702·73-s − 0.683·77-s − 0.450·79-s + 0.658·83-s − 0.108·85-s − 0.635·89-s − 0.615·95-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: \(0\)
Selberg data: \((2,\ 342720,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.204493049\)
\(L(\frac12)\) \(\approx\) \(3.204493049\)
\(L(\frac{3}{2})\) 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 \( 1 - 6 T + p T^{2} \)
13 \( 1 + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 4 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 14 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 18 T + p T^{2} \)
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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

−12.61446609283909, −12.17811086774887, −11.62324420347844, −11.19970549155022, −10.83732859708293, −10.25047235013970, −9.740951853725406, −9.425745180169014, −8.861089964763667, −8.644114751013835, −8.107027111682498, −7.265501610204407, −6.978035643601335, −6.488039337689007, −6.089718970678810, −5.749753778581374, −4.946359960579799, −4.346491999474514, −4.164148050255704, −3.422801108149326, −2.957351088996862, −2.243886615702185, −1.746708038305681, −1.124116979133130, −0.4968283932938655, 0.4968283932938655, 1.124116979133130, 1.746708038305681, 2.243886615702185, 2.957351088996862, 3.422801108149326, 4.164148050255704, 4.346491999474514, 4.946359960579799, 5.749753778581374, 6.089718970678810, 6.488039337689007, 6.978035643601335, 7.265501610204407, 8.107027111682498, 8.644114751013835, 8.861089964763667, 9.425745180169014, 9.740951853725406, 10.25047235013970, 10.83732859708293, 11.19970549155022, 11.62324420347844, 12.17811086774887, 12.61446609283909

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