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Properties

Label 2-129360-1.1-c1-0-132
Degree $2$
Conductor $129360$
Sign $-1$
Analytic cond. $1032.94$
Root an. cond. $32.1394$
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 + 11-s − 4·13-s + 15-s + 3·17-s + 7·19-s − 9·23-s + 25-s − 27-s + 3·29-s + 10·31-s − 33-s − 7·37-s + 4·39-s + 6·41-s + 43-s − 45-s + 3·47-s − 3·51-s + 6·53-s − 55-s − 7·57-s + 3·59-s + 8·61-s + 4·65-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s + 1/3·9-s + 0.301·11-s − 1.10·13-s + 0.258·15-s + 0.727·17-s + 1.60·19-s − 1.87·23-s + 1/5·25-s − 0.192·27-s + 0.557·29-s + 1.79·31-s − 0.174·33-s − 1.15·37-s + 0.640·39-s + 0.937·41-s + 0.152·43-s − 0.149·45-s + 0.437·47-s − 0.420·51-s + 0.824·53-s − 0.134·55-s − 0.927·57-s + 0.390·59-s + 1.02·61-s + 0.496·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 129360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 129360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(129360\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 7^{2} \cdot 11\)
Sign: $-1$
Analytic conductor: \(1032.94\)
Root analytic conductor: \(32.1394\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 129360,\ (\ :1/2),\ -1)\)

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 \( 1 \)
11 \( 1 - T \)
good13 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 - 7 T + p T^{2} \)
23 \( 1 + 9 T + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 - 10 T + p T^{2} \)
37 \( 1 + 7 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 - 3 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 3 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 - 3 T + p T^{2} \)
73 \( 1 + 16 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + 18 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 7 T + p T^{2} \)
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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

−13.88381023040019, −13.24380358193469, −12.50833011090392, −12.12220251415306, −11.89381159511911, −11.56626413616368, −10.88921741784229, −10.08968745180490, −9.970141030346993, −9.685471096915592, −8.697657270890788, −8.400124315763375, −7.626486180150938, −7.385357228178465, −6.920662122890320, −6.108896564652615, −5.804283879581423, −5.155597876642681, −4.686169725797929, −4.099962340468633, −3.577707055279154, −2.835718991160547, −2.326645062346635, −1.372137022866623, −0.8235283262029445, 0, 0.8235283262029445, 1.372137022866623, 2.326645062346635, 2.835718991160547, 3.577707055279154, 4.099962340468633, 4.686169725797929, 5.155597876642681, 5.804283879581423, 6.108896564652615, 6.920662122890320, 7.385357228178465, 7.626486180150938, 8.400124315763375, 8.697657270890788, 9.685471096915592, 9.970141030346993, 10.08968745180490, 10.88921741784229, 11.56626413616368, 11.89381159511911, 12.12220251415306, 12.50833011090392, 13.24380358193469, 13.88381023040019

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