Properties

Label 2-127050-1.1-c1-0-75
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 − 6·17-s − 18-s − 8·19-s − 21-s + 24-s − 2·26-s − 27-s + 28-s − 6·29-s − 4·31-s − 32-s + 6·34-s + 36-s + 10·37-s + 8·38-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 − 1.45·17-s − 0.235·18-s − 1.83·19-s − 0.218·21-s + 0.204·24-s − 0.392·26-s − 0.192·27-s + 0.188·28-s − 1.11·29-s − 0.718·31-s − 0.176·32-s + 1.02·34-s + 1/6·36-s + 1.64·37-s + 1.29·38-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 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
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

−13.45690842203659, −13.27969784859735, −12.72483443043265, −12.33013910296476, −11.55268169092743, −11.16019052699813, −10.93033945580563, −10.57250472423068, −9.811746997350457, −9.383039312472184, −8.774004537704545, −8.521477431535408, −7.852520044615784, −7.394933612564992, −6.701790664159168, −6.407809289720163, −5.866973315016840, −5.335890314314557, −4.484508945404868, −4.214441482339531, −3.584527883195541, −2.565021714726260, −2.122773246125257, −1.570031862089698, −0.6809653164119541, 0, 0.6809653164119541, 1.570031862089698, 2.122773246125257, 2.565021714726260, 3.584527883195541, 4.214441482339531, 4.484508945404868, 5.335890314314557, 5.866973315016840, 6.407809289720163, 6.701790664159168, 7.394933612564992, 7.852520044615784, 8.521477431535408, 8.774004537704545, 9.383039312472184, 9.811746997350457, 10.57250472423068, 10.93033945580563, 11.16019052699813, 11.55268169092743, 12.33013910296476, 12.72483443043265, 13.27969784859735, 13.45690842203659

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