Properties

Label 2-127995-1.1-c1-0-11
Degree $2$
Conductor $127995$
Sign $-1$
Analytic cond. $1022.04$
Root an. cond. $31.9694$
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 − 2·4-s − 5-s + 7-s + 9-s + 5·11-s + 2·12-s + 13-s + 15-s + 4·16-s + 3·17-s − 4·19-s + 2·20-s − 21-s − 23-s + 25-s − 27-s − 2·28-s − 4·29-s − 5·31-s − 5·33-s − 35-s − 2·36-s − 3·37-s − 39-s − 8·41-s − 8·43-s + ⋯
L(s)  = 1  − 0.577·3-s − 4-s − 0.447·5-s + 0.377·7-s + 1/3·9-s + 1.50·11-s + 0.577·12-s + 0.277·13-s + 0.258·15-s + 16-s + 0.727·17-s − 0.917·19-s + 0.447·20-s − 0.218·21-s − 0.208·23-s + 1/5·25-s − 0.192·27-s − 0.377·28-s − 0.742·29-s − 0.898·31-s − 0.870·33-s − 0.169·35-s − 1/3·36-s − 0.493·37-s − 0.160·39-s − 1.24·41-s − 1.21·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 127995 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 127995 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(127995\)    =    \(3 \cdot 5 \cdot 7 \cdot 23 \cdot 53\)
Sign: $-1$
Analytic conductor: \(1022.04\)
Root analytic conductor: \(31.9694\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 127995,\ (\ :1/2),\ -1)\)

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)$
bad3 \( 1 + T \)
5 \( 1 + T \)
7 \( 1 - T \)
23 \( 1 + T \)
53 \( 1 - T \)
good2 \( 1 + p T^{2} \)
11 \( 1 - 5 T + p T^{2} \)
13 \( 1 - T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 4 T + p T^{2} \)
31 \( 1 + 5 T + p T^{2} \)
37 \( 1 + 3 T + p T^{2} \)
41 \( 1 + 8 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 11 T + p T^{2} \)
61 \( 1 - 4 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 9 T + p T^{2} \)
73 \( 1 + 8 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 + 16 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 16 T + p T^{2} \)
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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

−13.83497871155745, −13.11360677781629, −12.83701132897691, −12.17916112008303, −11.90128800715693, −11.39175149447325, −10.94246555901745, −10.26530012165719, −9.935197957294385, −9.308847110229467, −8.854612764868703, −8.372883244292230, −8.045932458684218, −7.141489908214885, −6.917924642883081, −6.226494851706805, −5.532370087539548, −5.320573548462062, −4.533685423469794, −4.030529408979894, −3.755136391417493, −3.163639880134601, −1.957200228591649, −1.465497999328419, −0.7566730339660375, 0, 0.7566730339660375, 1.465497999328419, 1.957200228591649, 3.163639880134601, 3.755136391417493, 4.030529408979894, 4.533685423469794, 5.320573548462062, 5.532370087539548, 6.226494851706805, 6.917924642883081, 7.141489908214885, 8.045932458684218, 8.372883244292230, 8.854612764868703, 9.308847110229467, 9.935197957294385, 10.26530012165719, 10.94246555901745, 11.39175149447325, 11.90128800715693, 12.17916112008303, 12.83701132897691, 13.11360677781629, 13.83497871155745

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