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Properties

Label 2-364560-1.1-c1-0-20
Degree $2$
Conductor $364560$
Sign $-1$
Analytic cond. $2911.02$
Root an. cond. $53.9539$
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  − 3-s − 5-s + 9-s − 3·11-s + 2·13-s + 15-s − 8·17-s − 7·19-s − 7·23-s + 25-s − 27-s − 8·29-s − 31-s + 3·33-s − 4·37-s − 2·39-s − 43-s − 45-s + 6·47-s + 8·51-s + 5·53-s + 3·55-s + 7·57-s + 6·59-s − 2·61-s − 2·65-s − 10·67-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s + 1/3·9-s − 0.904·11-s + 0.554·13-s + 0.258·15-s − 1.94·17-s − 1.60·19-s − 1.45·23-s + 1/5·25-s − 0.192·27-s − 1.48·29-s − 0.179·31-s + 0.522·33-s − 0.657·37-s − 0.320·39-s − 0.152·43-s − 0.149·45-s + 0.875·47-s + 1.12·51-s + 0.686·53-s + 0.404·55-s + 0.927·57-s + 0.781·59-s − 0.256·61-s − 0.248·65-s − 1.22·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 364560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 364560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(364560\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 7^{2} \cdot 31\)
Sign: $-1$
Analytic conductor: \(2911.02\)
Root analytic conductor: \(53.9539\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 364560,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
5 \( 1 + T \)
7 \( 1 \)
31 \( 1 + T \)
good11 \( 1 + 3 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 8 T + p T^{2} \)
19 \( 1 + 7 T + p T^{2} \)
23 \( 1 + 7 T + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 - 5 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 10 T + p T^{2} \)
71 \( 1 + 9 T + p T^{2} \)
73 \( 1 + T + p T^{2} \)
79 \( 1 + 13 T + p T^{2} \)
83 \( 1 + 16 T + p T^{2} \)
89 \( 1 - 3 T + p T^{2} \)
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

−12.75412279307930, −12.34450838332423, −11.67375881190509, −11.40482237913800, −10.93011894440215, −10.52443113590211, −10.28151150945327, −9.623015613719979, −8.937899133298637, −8.579974212632428, −8.359825714526882, −7.514207911660322, −7.304914566396810, −6.682397071377852, −6.237963610403472, −5.737875054034218, −5.413501666492376, −4.545932236800173, −4.257167484795205, −3.977525437067600, −3.204799290492023, −2.442376691190794, −2.032091859182830, −1.532220632469576, −0.3761411843571645, 0, 0.3761411843571645, 1.532220632469576, 2.032091859182830, 2.442376691190794, 3.204799290492023, 3.977525437067600, 4.257167484795205, 4.545932236800173, 5.413501666492376, 5.737875054034218, 6.237963610403472, 6.682397071377852, 7.304914566396810, 7.514207911660322, 8.359825714526882, 8.579974212632428, 8.937899133298637, 9.623015613719979, 10.28151150945327, 10.52443113590211, 10.93011894440215, 11.40482237913800, 11.67375881190509, 12.34450838332423, 12.75412279307930

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