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Properties

Label 2-95550-1.1-c1-0-274
Degree $2$
Conductor $95550$
Sign $1$
Analytic cond. $762.970$
Root an. cond. $27.6219$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $2$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3-s + 4-s + 6-s − 8-s + 9-s − 5·11-s − 12-s − 13-s + 16-s + 7·17-s − 18-s − 7·19-s + 5·22-s − 2·23-s + 24-s + 26-s − 27-s − 9·29-s − 32-s + 5·33-s − 7·34-s + 36-s − 4·37-s + 7·38-s + 39-s − 4·41-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.353·8-s + 1/3·9-s − 1.50·11-s − 0.288·12-s − 0.277·13-s + 1/4·16-s + 1.69·17-s − 0.235·18-s − 1.60·19-s + 1.06·22-s − 0.417·23-s + 0.204·24-s + 0.196·26-s − 0.192·27-s − 1.67·29-s − 0.176·32-s + 0.870·33-s − 1.20·34-s + 1/6·36-s − 0.657·37-s + 1.13·38-s + 0.160·39-s − 0.624·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 95550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(95550\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7^{2} \cdot 13\)
Sign: $1$
Analytic conductor: \(762.970\)
Root analytic conductor: \(27.6219\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((2,\ 95550,\ (\ :1/2),\ 1)\)

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 \( 1 \)
13 \( 1 + T \)
good11 \( 1 + 5 T + p T^{2} \)
17 \( 1 - 7 T + p T^{2} \)
19 \( 1 + 7 T + p T^{2} \)
23 \( 1 + 2 T + p T^{2} \)
29 \( 1 + 9 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 + 4 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 + 3 T + p T^{2} \)
53 \( 1 + T + p T^{2} \)
59 \( 1 + 7 T + p T^{2} \)
61 \( 1 + 13 T + p T^{2} \)
67 \( 1 + 3 T + p T^{2} \)
71 \( 1 - 9 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 14 T + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
89 \( 1 - 12 T + p T^{2} \)
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

−14.46990713500269, −13.68615901967446, −13.19080337013572, −12.72925274508044, −12.23664909583942, −11.91243062947392, −11.18468248631323, −10.65255531259773, −10.43522355817548, −10.00122408202085, −9.360398082993278, −8.895318358501528, −8.098799050155311, −7.777254693502424, −7.498078813623835, −6.704431473171367, −6.141994417202478, −5.707612361029146, −5.077784338505680, −4.723472176087741, −3.684353621425950, −3.323422689009489, −2.399691870588382, −1.940961446346798, −1.187081295951151, 0, 0, 1.187081295951151, 1.940961446346798, 2.399691870588382, 3.323422689009489, 3.684353621425950, 4.723472176087741, 5.077784338505680, 5.707612361029146, 6.141994417202478, 6.704431473171367, 7.498078813623835, 7.777254693502424, 8.098799050155311, 8.895318358501528, 9.360398082993278, 10.00122408202085, 10.43522355817548, 10.65255531259773, 11.18468248631323, 11.91243062947392, 12.23664909583942, 12.72925274508044, 13.19080337013572, 13.68615901967446, 14.46990713500269

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