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Properties

Label 2-342720-1.1-c1-0-266
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 + 2·13-s − 17-s − 4·19-s + 2·23-s + 25-s + 2·29-s + 8·31-s + 35-s − 6·37-s + 10·41-s + 2·47-s + 49-s + 10·53-s + 2·55-s + 6·59-s + 10·61-s − 2·65-s − 8·67-s + 8·71-s + 10·73-s + 2·77-s + 6·79-s − 8·83-s + 85-s + ⋯
L(s)  = 1  − 0.447·5-s − 0.377·7-s − 0.603·11-s + 0.554·13-s − 0.242·17-s − 0.917·19-s + 0.417·23-s + 1/5·25-s + 0.371·29-s + 1.43·31-s + 0.169·35-s − 0.986·37-s + 1.56·41-s + 0.291·47-s + 1/7·49-s + 1.37·53-s + 0.269·55-s + 0.781·59-s + 1.28·61-s − 0.248·65-s − 0.977·67-s + 0.949·71-s + 1.17·73-s + 0.227·77-s + 0.675·79-s − 0.878·83-s + 0.108·85-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 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 2 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 - 2 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 6 T + p T^{2} \)
83 \( 1 + 8 T + p T^{2} \)
89 \( 1 + 4 T + p T^{2} \)
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

−12.81580999799947, −12.19638169152584, −12.11093072581457, −11.35195345066751, −10.83922417441285, −10.73486146217885, −10.06816993360626, −9.674225751556925, −9.106708175305176, −8.501361428299661, −8.339574254477495, −7.839763383263991, −7.160614181772901, −6.714729267819722, −6.478943672642262, −5.592716573849327, −5.487537273201617, −4.655370350501450, −4.209467948646684, −3.823232525757410, −3.164065634251962, −2.534477405121963, −2.282988851370624, −1.260898684979677, −0.7446744132735612, 0, 0.7446744132735612, 1.260898684979677, 2.282988851370624, 2.534477405121963, 3.164065634251962, 3.823232525757410, 4.209467948646684, 4.655370350501450, 5.487537273201617, 5.592716573849327, 6.478943672642262, 6.714729267819722, 7.160614181772901, 7.839763383263991, 8.339574254477495, 8.501361428299661, 9.106708175305176, 9.674225751556925, 10.06816993360626, 10.73486146217885, 10.83922417441285, 11.35195345066751, 12.11093072581457, 12.19638169152584, 12.81580999799947

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