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Properties

Label 2-342720-1.1-c1-0-351
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 + 4·11-s + 6·13-s − 17-s − 4·19-s − 4·23-s + 25-s + 2·29-s + 4·31-s + 35-s + 6·37-s + 2·41-s + 8·47-s + 49-s − 10·53-s − 4·55-s + 8·59-s + 6·61-s − 6·65-s + 6·73-s − 4·77-s − 12·79-s + 12·83-s + 85-s + 6·89-s − 6·91-s + ⋯
L(s)  = 1  − 0.447·5-s − 0.377·7-s + 1.20·11-s + 1.66·13-s − 0.242·17-s − 0.917·19-s − 0.834·23-s + 1/5·25-s + 0.371·29-s + 0.718·31-s + 0.169·35-s + 0.986·37-s + 0.312·41-s + 1.16·47-s + 1/7·49-s − 1.37·53-s − 0.539·55-s + 1.04·59-s + 0.768·61-s − 0.744·65-s + 0.702·73-s − 0.455·77-s − 1.35·79-s + 1.31·83-s + 0.108·85-s + 0.635·89-s − 0.628·91-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 \( 1 - 4 T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + 12 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
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

−12.82132705001531, −12.26071969514899, −11.88946651595309, −11.43897667274638, −10.99981026974807, −10.64932986511185, −10.08032672023142, −9.553766819439944, −9.085684266072301, −8.685938020311287, −8.227789015112244, −7.902503726469120, −7.169658050583129, −6.608493948371904, −6.299130640902052, −6.034145385472680, −5.347298473956777, −4.567657710052374, −4.081115237320084, −3.879904447543518, −3.331296630192481, −2.609439506521553, −2.056762147644640, −1.234711494183646, −0.8993562152960191, 0, 0.8993562152960191, 1.234711494183646, 2.056762147644640, 2.609439506521553, 3.331296630192481, 3.879904447543518, 4.081115237320084, 4.567657710052374, 5.347298473956777, 6.034145385472680, 6.299130640902052, 6.608493948371904, 7.169658050583129, 7.902503726469120, 8.227789015112244, 8.685938020311287, 9.085684266072301, 9.553766819439944, 10.08032672023142, 10.64932986511185, 10.99981026974807, 11.43897667274638, 11.88946651595309, 12.26071969514899, 12.82132705001531

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