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Properties

Label 2-95550-1.1-c1-0-111
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3-s + 4-s + 6-s − 8-s + 9-s − 2·11-s − 12-s − 13-s + 16-s − 2·17-s − 18-s − 4·19-s + 2·22-s + 6·23-s + 24-s + 26-s − 27-s + 2·29-s − 4·31-s − 32-s + 2·33-s + 2·34-s + 36-s + 2·37-s + 4·38-s + 39-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 − 0.603·11-s − 0.288·12-s − 0.277·13-s + 1/4·16-s − 0.485·17-s − 0.235·18-s − 0.917·19-s + 0.426·22-s + 1.25·23-s + 0.204·24-s + 0.196·26-s − 0.192·27-s + 0.371·29-s − 0.718·31-s − 0.176·32-s + 0.348·33-s + 0.342·34-s + 1/6·36-s + 0.328·37-s + 0.648·38-s + 0.160·39-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: \(1\)
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 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 6 T + p T^{2} \)
47 \( 1 - 2 T + p T^{2} \)
53 \( 1 - 4 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 8 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 + 8 T + p T^{2} \)
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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

−14.07099840234884, −13.40531542980526, −12.90582076179807, −12.62104836187342, −12.03761308407405, −11.38039918665259, −11.07483789415483, −10.57505596719141, −10.23913616507262, −9.547529157873659, −9.119460112230815, −8.601390010388454, −8.049535278188933, −7.512482519337269, −6.932082519705781, −6.617014136356220, −5.896041037146442, −5.447775719878349, −4.763608609777056, −4.332533739861291, −3.500795092315739, −2.780851479882308, −2.245079465859825, −1.518662935857386, −0.7107454764448823, 0, 0.7107454764448823, 1.518662935857386, 2.245079465859825, 2.780851479882308, 3.500795092315739, 4.332533739861291, 4.763608609777056, 5.447775719878349, 5.896041037146442, 6.617014136356220, 6.932082519705781, 7.512482519337269, 8.049535278188933, 8.601390010388454, 9.119460112230815, 9.547529157873659, 10.23913616507262, 10.57505596719141, 11.07483789415483, 11.38039918665259, 12.03761308407405, 12.62104836187342, 12.90582076179807, 13.40531542980526, 14.07099840234884

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