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Properties

Label 2-129360-1.1-c1-0-123
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 − 5·13-s − 15-s − 6·17-s + 2·19-s + 6·23-s + 25-s + 27-s − 7·31-s + 33-s + 2·37-s − 5·39-s + 43-s − 45-s + 6·47-s − 6·51-s + 6·53-s − 55-s + 2·57-s − 3·59-s + 10·61-s + 5·65-s − 2·67-s + 6·69-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s + 1/3·9-s + 0.301·11-s − 1.38·13-s − 0.258·15-s − 1.45·17-s + 0.458·19-s + 1.25·23-s + 1/5·25-s + 0.192·27-s − 1.25·31-s + 0.174·33-s + 0.328·37-s − 0.800·39-s + 0.152·43-s − 0.149·45-s + 0.875·47-s − 0.840·51-s + 0.824·53-s − 0.134·55-s + 0.264·57-s − 0.390·59-s + 1.28·61-s + 0.620·65-s − 0.244·67-s + 0.722·69-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 \( 1 + 5 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 7 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 3 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 + 15 T + p T^{2} \)
73 \( 1 + 11 T + p T^{2} \)
79 \( 1 + 14 T + p T^{2} \)
83 \( 1 - 3 T + p T^{2} \)
89 \( 1 - 3 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−13.68573587637178, −13.20296725490924, −12.85312854396578, −12.33946115758803, −11.81380120853530, −11.27875871452537, −10.98603937755637, −10.18157381338115, −9.931932892768191, −9.128669299978097, −8.889403892919587, −8.580664401907521, −7.601144091785830, −7.345279791112203, −7.070167287420064, −6.389463640289476, −5.658872841132542, −5.088897135313311, −4.479622403390323, −4.185491978581864, −3.397113177857420, −2.851991635856022, −2.327063129658217, −1.705958439036725, −0.8059137933496975, 0, 0.8059137933496975, 1.705958439036725, 2.327063129658217, 2.851991635856022, 3.397113177857420, 4.185491978581864, 4.479622403390323, 5.088897135313311, 5.658872841132542, 6.389463640289476, 7.070167287420064, 7.345279791112203, 7.601144091785830, 8.580664401907521, 8.889403892919587, 9.128669299978097, 9.931932892768191, 10.18157381338115, 10.98603937755637, 11.27875871452537, 11.81380120853530, 12.33946115758803, 12.85312854396578, 13.20296725490924, 13.68573587637178

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