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Properties

Label 2-342720-1.1-c1-0-85
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 + 6·13-s + 17-s − 4·19-s + 25-s − 2·29-s − 35-s + 6·37-s + 10·41-s − 4·43-s − 4·47-s + 49-s + 6·53-s + 4·59-s + 14·61-s − 6·65-s + 12·67-s − 12·71-s − 10·73-s − 12·79-s + 16·83-s − 85-s + 6·89-s + 6·91-s + 4·95-s − 18·97-s + ⋯
L(s)  = 1  − 0.447·5-s + 0.377·7-s + 1.66·13-s + 0.242·17-s − 0.917·19-s + 1/5·25-s − 0.371·29-s − 0.169·35-s + 0.986·37-s + 1.56·41-s − 0.609·43-s − 0.583·47-s + 1/7·49-s + 0.824·53-s + 0.520·59-s + 1.79·61-s − 0.744·65-s + 1.46·67-s − 1.42·71-s − 1.17·73-s − 1.35·79-s + 1.75·83-s − 0.108·85-s + 0.635·89-s + 0.628·91-s + 0.410·95-s − 1.82·97-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

\(L(1)\) \(\approx\) \(2.433321400\)
\(L(\frac12)\) \(\approx\) \(2.433321400\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 12 T + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
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

−12.73760686821937, −11.86701333759244, −11.75285053767432, −11.15917923899661, −10.74990525998055, −10.57476445091446, −9.732477823856231, −9.435732557707990, −8.721322019002775, −8.462848280924555, −8.042907815328133, −7.629358748846621, −6.914430344989309, −6.576854573480423, −6.036320416715923, −5.529591339431707, −5.128657966863688, −4.270335283135249, −3.988100394100112, −3.709076969397482, −2.771547289169361, −2.485242570990494, −1.535751407142386, −1.196345099030934, −0.4321847566218038, 0.4321847566218038, 1.196345099030934, 1.535751407142386, 2.485242570990494, 2.771547289169361, 3.709076969397482, 3.988100394100112, 4.270335283135249, 5.128657966863688, 5.529591339431707, 6.036320416715923, 6.576854573480423, 6.914430344989309, 7.629358748846621, 8.042907815328133, 8.462848280924555, 8.721322019002775, 9.435732557707990, 9.732477823856231, 10.57476445091446, 10.74990525998055, 11.15917923899661, 11.75285053767432, 11.86701333759244, 12.73760686821937

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