Properties

Label 2-190575-1.1-c1-0-107
Degree $2$
Conductor $190575$
Sign $-1$
Analytic cond. $1521.74$
Root an. cond. $39.0096$
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 − 4-s + 7-s − 3·8-s + 4·13-s + 14-s − 16-s + 4·19-s − 6·23-s + 4·26-s − 28-s + 6·31-s + 5·32-s − 2·37-s + 4·38-s + 2·41-s − 12·43-s − 6·46-s + 8·47-s + 49-s − 4·52-s − 14·53-s − 3·56-s − 6·59-s + 14·61-s + 6·62-s + 7·64-s + ⋯
L(s)  = 1  + 0.707·2-s − 1/2·4-s + 0.377·7-s − 1.06·8-s + 1.10·13-s + 0.267·14-s − 1/4·16-s + 0.917·19-s − 1.25·23-s + 0.784·26-s − 0.188·28-s + 1.07·31-s + 0.883·32-s − 0.328·37-s + 0.648·38-s + 0.312·41-s − 1.82·43-s − 0.884·46-s + 1.16·47-s + 1/7·49-s − 0.554·52-s − 1.92·53-s − 0.400·56-s − 0.781·59-s + 1.79·61-s + 0.762·62-s + 7/8·64-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 190575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 190575 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(190575\)    =    \(3^{2} \cdot 5^{2} \cdot 7 \cdot 11^{2}\)
Sign: $-1$
Analytic conductor: \(1521.74\)
Root analytic conductor: \(39.0096\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 190575,\ (\ :1/2),\ -1)\)

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 \( 1 \)
5 \( 1 \)
7 \( 1 - T \)
11 \( 1 \)
good2 \( 1 - T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 6 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 14 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 - 10 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
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

−13.37253515328346, −13.07079858235694, −12.28597543916955, −12.13094322401124, −11.48637438353808, −11.24386259333891, −10.49132659206046, −9.993671819649953, −9.636557166278189, −9.030585450643326, −8.504342692362243, −8.187092915494398, −7.711259540558472, −6.983840131233791, −6.312081285870233, −6.092202282811523, −5.380557225016283, −5.091895191480936, −4.374581946823756, −4.041921604493336, −3.391343615238745, −3.043085655586491, −2.224177428084808, −1.487372282547842, −0.8646654880796494, 0, 0.8646654880796494, 1.487372282547842, 2.224177428084808, 3.043085655586491, 3.391343615238745, 4.041921604493336, 4.374581946823756, 5.091895191480936, 5.380557225016283, 6.092202282811523, 6.312081285870233, 6.983840131233791, 7.711259540558472, 8.187092915494398, 8.504342692362243, 9.030585450643326, 9.636557166278189, 9.993671819649953, 10.49132659206046, 11.24386259333891, 11.48637438353808, 12.13094322401124, 12.28597543916955, 13.07079858235694, 13.37253515328346

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