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Properties

Label 2-95550-1.1-c1-0-178
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 + 4·11-s + 12-s + 13-s + 16-s − 4·17-s − 18-s − 7·19-s − 4·22-s − 4·23-s − 24-s − 26-s + 27-s + 5·29-s − 4·31-s − 32-s + 4·33-s + 4·34-s + 36-s − 9·37-s + 7·38-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 + 1.20·11-s + 0.288·12-s + 0.277·13-s + 1/4·16-s − 0.970·17-s − 0.235·18-s − 1.60·19-s − 0.852·22-s − 0.834·23-s − 0.204·24-s − 0.196·26-s + 0.192·27-s + 0.928·29-s − 0.718·31-s − 0.176·32-s + 0.696·33-s + 0.685·34-s + 1/6·36-s − 1.47·37-s + 1.13·38-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 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 + 7 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 5 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 9 T + p T^{2} \)
41 \( 1 - 5 T + p T^{2} \)
43 \( 1 - 10 T + p T^{2} \)
47 \( 1 - 3 T + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 + 4 T + p T^{2} \)
67 \( 1 - 7 T + p T^{2} \)
71 \( 1 + 15 T + p T^{2} \)
73 \( 1 - 12 T + p T^{2} \)
79 \( 1 - 7 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 + 16 T + p T^{2} \)
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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

−14.08276440892375, −13.74057701131084, −12.98811052281918, −12.41259823996029, −12.27414319944867, −11.41966846820498, −11.02111076049387, −10.56698849418307, −10.08886327435841, −9.338762674719759, −9.095325573382152, −8.619620855368014, −8.213532141135026, −7.620409003565459, −6.926031930554002, −6.572231642163912, −6.147516942589569, −5.455311298808324, −4.460777191789938, −4.162519012579226, −3.606278198087301, −2.811201878343043, −2.107373892844290, −1.760464608574863, −0.8989888374650802, 0, 0.8989888374650802, 1.760464608574863, 2.107373892844290, 2.811201878343043, 3.606278198087301, 4.162519012579226, 4.460777191789938, 5.455311298808324, 6.147516942589569, 6.572231642163912, 6.926031930554002, 7.620409003565459, 8.213532141135026, 8.619620855368014, 9.095325573382152, 9.338762674719759, 10.08886327435841, 10.56698849418307, 11.02111076049387, 11.41966846820498, 12.27414319944867, 12.41259823996029, 12.98811052281918, 13.74057701131084, 14.08276440892375

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