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Properties

Label 2-127050-1.1-c1-0-221
Degree $2$
Conductor $127050$
Sign $-1$
Analytic cond. $1014.49$
Root an. cond. $31.8512$
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  + 2-s + 3-s + 4-s + 6-s + 7-s + 8-s + 9-s + 12-s − 2·13-s + 14-s + 16-s − 17-s + 18-s + 19-s + 21-s + 4·23-s + 24-s − 2·26-s + 27-s + 28-s − 10·31-s + 32-s − 34-s + 36-s + 2·37-s + 38-s − 2·39-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s + 0.288·12-s − 0.554·13-s + 0.267·14-s + 1/4·16-s − 0.242·17-s + 0.235·18-s + 0.229·19-s + 0.218·21-s + 0.834·23-s + 0.204·24-s − 0.392·26-s + 0.192·27-s + 0.188·28-s − 1.79·31-s + 0.176·32-s − 0.171·34-s + 1/6·36-s + 0.328·37-s + 0.162·38-s − 0.320·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 127050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 127050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(127050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7 \cdot 11^{2}\)
Sign: $-1$
Analytic conductor: \(1014.49\)
Root analytic conductor: \(31.8512\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 127050,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
3 \( 1 - T \)
5 \( 1 \)
7 \( 1 - T \)
11 \( 1 \)
good13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + T + p T^{2} \)
19 \( 1 - T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 10 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 4 T + p T^{2} \)
43 \( 1 - 7 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 5 T + p T^{2} \)
59 \( 1 - 9 T + p T^{2} \)
61 \( 1 + 13 T + p T^{2} \)
67 \( 1 + 5 T + p T^{2} \)
71 \( 1 + 13 T + p T^{2} \)
73 \( 1 + 7 T + p T^{2} \)
79 \( 1 - 5 T + p T^{2} \)
83 \( 1 + 16 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 + 13 T + p T^{2} \)
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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

−13.77222019810845, −13.21192947755019, −12.87515279013779, −12.48210243417255, −11.88609191621214, −11.33528588085578, −10.96505131161514, −10.48342326704155, −9.827436637682936, −9.334620337514368, −8.899444627334235, −8.348946731492515, −7.670982223998357, −7.293564691311497, −6.999869781030101, −6.114895839641903, −5.739138084069627, −5.091380288341116, −4.533981193802874, −4.207574007888388, −3.340803606343081, −3.044780930467227, −2.308776494799250, −1.770386129090671, −1.095703290226594, 0, 1.095703290226594, 1.770386129090671, 2.308776494799250, 3.044780930467227, 3.340803606343081, 4.207574007888388, 4.533981193802874, 5.091380288341116, 5.739138084069627, 6.114895839641903, 6.999869781030101, 7.293564691311497, 7.670982223998357, 8.348946731492515, 8.899444627334235, 9.334620337514368, 9.827436637682936, 10.48342326704155, 10.96505131161514, 11.33528588085578, 11.88609191621214, 12.48210243417255, 12.87515279013779, 13.21192947755019, 13.77222019810845

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