Properties

Label 2-10725-1.1-c1-0-4
Degree $2$
Conductor $10725$
Sign $-1$
Analytic cond. $85.6395$
Root an. cond. $9.25416$
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 − 3·8-s + 9-s − 11-s + 12-s − 13-s − 16-s + 6·17-s + 18-s − 4·19-s − 22-s + 8·23-s + 3·24-s − 26-s − 27-s − 10·29-s + 5·32-s + 33-s + 6·34-s − 36-s − 6·37-s − 4·38-s + 39-s + 10·41-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s − 1/2·4-s − 0.408·6-s − 1.06·8-s + 1/3·9-s − 0.301·11-s + 0.288·12-s − 0.277·13-s − 1/4·16-s + 1.45·17-s + 0.235·18-s − 0.917·19-s − 0.213·22-s + 1.66·23-s + 0.612·24-s − 0.196·26-s − 0.192·27-s − 1.85·29-s + 0.883·32-s + 0.174·33-s + 1.02·34-s − 1/6·36-s − 0.986·37-s − 0.648·38-s + 0.160·39-s + 1.56·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10725 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10725 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(10725\)    =    \(3 \cdot 5^{2} \cdot 11 \cdot 13\)
Sign: $-1$
Analytic conductor: \(85.6395\)
Root analytic conductor: \(9.25416\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 10725,\ (\ :1/2),\ -1)\)

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 \( 1 \)
11 \( 1 + T \)
13 \( 1 + T \)
good2 \( 1 - T + p T^{2} \)
7 \( 1 + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 + 10 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
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 \( 1 - 14 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
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

−16.81045556593852, −16.40755975045793, −15.46376350524279, −15.01981980507851, −14.54305285927140, −14.03205709519323, −13.15375462715251, −12.79558165457163, −12.52533849932577, −11.63347644335024, −11.19844699977157, −10.47438568683167, −9.778984947688950, −9.299058590354278, −8.564270532800161, −7.850014270711203, −7.139516450722529, −6.431797028813529, −5.642860742416598, −5.234625967618866, −4.696128610850488, −3.760984818940426, −3.301398275251033, −2.261846230600368, −1.064159688911846, 0, 1.064159688911846, 2.261846230600368, 3.301398275251033, 3.760984818940426, 4.696128610850488, 5.234625967618866, 5.642860742416598, 6.431797028813529, 7.139516450722529, 7.850014270711203, 8.564270532800161, 9.299058590354278, 9.778984947688950, 10.47438568683167, 11.19844699977157, 11.63347644335024, 12.52533849932577, 12.79558165457163, 13.15375462715251, 14.03205709519323, 14.54305285927140, 15.01981980507851, 15.46376350524279, 16.40755975045793, 16.81045556593852

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