Properties

Label 2-54150-1.1-c1-0-18
Degree $2$
Conductor $54150$
Sign $1$
Analytic cond. $432.389$
Root an. cond. $20.7939$
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 + 3-s + 4-s − 6-s − 2·7-s − 8-s + 9-s + 12-s + 2·13-s + 2·14-s + 16-s + 6·17-s − 18-s − 2·21-s − 6·23-s − 24-s − 2·26-s + 27-s − 2·28-s + 4·29-s − 32-s − 6·34-s + 36-s + 10·37-s + 2·39-s + 8·41-s + 2·42-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.755·7-s − 0.353·8-s + 1/3·9-s + 0.288·12-s + 0.554·13-s + 0.534·14-s + 1/4·16-s + 1.45·17-s − 0.235·18-s − 0.436·21-s − 1.25·23-s − 0.204·24-s − 0.392·26-s + 0.192·27-s − 0.377·28-s + 0.742·29-s − 0.176·32-s − 1.02·34-s + 1/6·36-s + 1.64·37-s + 0.320·39-s + 1.24·41-s + 0.308·42-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 54150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(54150\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(432.389\)
Root analytic conductor: \(20.7939\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{54150} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 54150,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.198111827\)
\(L(\frac12)\) \(\approx\) \(2.198111827\)
\(L(\frac{3}{2})\) 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 \( 1 \)
19 \( 1 \)
good7 \( 1 + 2 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 4 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 8 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 - 2 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 2 T + p T^{2} \)
89 \( 1 - 8 T + p T^{2} \)
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

−14.44987595689119, −14.10957185074011, −13.21497317664114, −13.07386926239970, −12.27307519076226, −11.96586304477372, −11.30059265806240, −10.68948563167616, −10.07687008136090, −9.769594058058828, −9.380880940385910, −8.651797026249202, −8.183959601426114, −7.738934460258048, −7.243364457880370, −6.473445547739808, −6.047978967560342, −5.560159167554416, −4.582159173304203, −3.909697290357201, −3.348600448222001, −2.774235491531735, −2.121503023408737, −1.235344989979892, −0.6083823631895613, 0.6083823631895613, 1.235344989979892, 2.121503023408737, 2.774235491531735, 3.348600448222001, 3.909697290357201, 4.582159173304203, 5.560159167554416, 6.047978967560342, 6.473445547739808, 7.243364457880370, 7.738934460258048, 8.183959601426114, 8.651797026249202, 9.380880940385910, 9.769594058058828, 10.07687008136090, 10.68948563167616, 11.30059265806240, 11.96586304477372, 12.27307519076226, 13.07386926239970, 13.21497317664114, 14.10957185074011, 14.44987595689119

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