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Properties

Label 2-129360-1.1-c1-0-137
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 + 2·13-s + 15-s + 6·17-s − 4·19-s − 8·23-s + 25-s − 27-s + 6·29-s + 8·31-s + 33-s + 6·37-s − 2·39-s + 6·41-s + 4·43-s − 45-s − 8·47-s − 6·51-s − 10·53-s + 55-s + 4·57-s − 12·59-s + 10·61-s − 2·65-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s + 1/3·9-s − 0.301·11-s + 0.554·13-s + 0.258·15-s + 1.45·17-s − 0.917·19-s − 1.66·23-s + 1/5·25-s − 0.192·27-s + 1.11·29-s + 1.43·31-s + 0.174·33-s + 0.986·37-s − 0.320·39-s + 0.937·41-s + 0.609·43-s − 0.149·45-s − 1.16·47-s − 0.840·51-s − 1.37·53-s + 0.134·55-s + 0.529·57-s − 1.56·59-s + 1.28·61-s − 0.248·65-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 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−13.90855129524862, −13.01365200353652, −12.73374422242965, −12.23892931619138, −11.89554596669352, −11.20299864321671, −11.04779139553918, −10.26280349520639, −9.918572071319142, −9.637596476004948, −8.645284601038723, −8.307161743066967, −7.773445406469836, −7.550939791218176, −6.495250198879026, −6.306369746756814, −5.904639591657543, −5.090858866559873, −4.695168486551408, −4.037380081799387, −3.643921901412776, −2.841209126498244, −2.304774645478857, −1.365957494778239, −0.8401707311122338, 0, 0.8401707311122338, 1.365957494778239, 2.304774645478857, 2.841209126498244, 3.643921901412776, 4.037380081799387, 4.695168486551408, 5.090858866559873, 5.904639591657543, 6.306369746756814, 6.495250198879026, 7.550939791218176, 7.773445406469836, 8.307161743066967, 8.645284601038723, 9.637596476004948, 9.918572071319142, 10.26280349520639, 11.04779139553918, 11.20299864321671, 11.89554596669352, 12.23892931619138, 12.73374422242965, 13.01365200353652, 13.90855129524862

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