Properties

Label 2-45675-1.1-c1-0-3
Degree $2$
Conductor $45675$
Sign $1$
Analytic cond. $364.716$
Root an. cond. $19.0975$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 4-s − 7-s + 3·8-s − 4·11-s + 2·13-s + 14-s − 16-s + 2·17-s − 4·19-s + 4·22-s − 2·26-s + 28-s − 29-s − 8·31-s − 5·32-s − 2·34-s + 10·37-s + 4·38-s + 6·41-s − 12·43-s + 4·44-s − 8·47-s + 49-s − 2·52-s + 6·53-s − 3·56-s + ⋯
L(s)  = 1  − 0.707·2-s − 1/2·4-s − 0.377·7-s + 1.06·8-s − 1.20·11-s + 0.554·13-s + 0.267·14-s − 1/4·16-s + 0.485·17-s − 0.917·19-s + 0.852·22-s − 0.392·26-s + 0.188·28-s − 0.185·29-s − 1.43·31-s − 0.883·32-s − 0.342·34-s + 1.64·37-s + 0.648·38-s + 0.937·41-s − 1.82·43-s + 0.603·44-s − 1.16·47-s + 1/7·49-s − 0.277·52-s + 0.824·53-s − 0.400·56-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 45675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(45675\)    =    \(3^{2} \cdot 5^{2} \cdot 7 \cdot 29\)
Sign: $1$
Analytic conductor: \(364.716\)
Root analytic conductor: \(19.0975\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 45675,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4636654744\)
\(L(\frac12)\) \(\approx\) \(0.4636654744\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
29 \( 1 + T \)
good2 \( 1 + T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 16 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 6 T + p T^{2} \)
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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

−14.69873332352908, −14.09043300974672, −13.39384978481490, −13.14722620589518, −12.72634779254843, −12.16282010434815, −11.17968662918982, −10.93754803503441, −10.42545105362292, −9.789154417598280, −9.480746745535211, −8.828298439266968, −8.305242773491981, −7.781002863305730, −7.493413237800720, −6.546518920003853, −6.110546149579394, −5.236710873112097, −4.974040866063541, −4.068935007049539, −3.610222015340557, −2.783951963282679, −2.024989216745849, −1.241343533690278, −0.2945948276081118, 0.2945948276081118, 1.241343533690278, 2.024989216745849, 2.783951963282679, 3.610222015340557, 4.068935007049539, 4.974040866063541, 5.236710873112097, 6.110546149579394, 6.546518920003853, 7.493413237800720, 7.781002863305730, 8.305242773491981, 8.828298439266968, 9.480746745535211, 9.789154417598280, 10.42545105362292, 10.93754803503441, 11.17968662918982, 12.16282010434815, 12.72634779254843, 13.14722620589518, 13.39384978481490, 14.09043300974672, 14.69873332352908

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