Properties

Label 2-95550-1.1-c1-0-54
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 $0$

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 + 8·17-s − 18-s + 6·19-s − 4·22-s − 6·23-s + 24-s + 26-s − 27-s − 4·29-s − 32-s − 4·33-s − 8·34-s + 36-s + 2·37-s − 6·38-s + 39-s + 2·41-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 + 1.20·11-s − 0.288·12-s − 0.277·13-s + 1/4·16-s + 1.94·17-s − 0.235·18-s + 1.37·19-s − 0.852·22-s − 1.25·23-s + 0.204·24-s + 0.196·26-s − 0.192·27-s − 0.742·29-s − 0.176·32-s − 0.696·33-s − 1.37·34-s + 1/6·36-s + 0.328·37-s − 0.973·38-s + 0.160·39-s + 0.312·41-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: $\chi_{95550} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 95550,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.785388416\)
\(L(\frac12)\) \(\approx\) \(1.785388416\)
\(L(\frac{3}{2})\) 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 \( 1 - 8 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 4 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 8 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
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

−13.84794995635352, −13.33609257513024, −12.57972610085808, −12.03811578497620, −11.80353524638011, −11.59959800722575, −10.67745184781601, −10.34181303107336, −9.797309320659902, −9.379185072505910, −9.057193474583628, −8.171832963629551, −7.689786858108362, −7.401793990800578, −6.769556199789096, −6.137942597437229, −5.637224630071349, −5.346728010207325, −4.408040460557867, −3.800698299305128, −3.335431208620378, −2.548759464057457, −1.669334634032827, −1.166073717451803, −0.5635371654678414, 0.5635371654678414, 1.166073717451803, 1.669334634032827, 2.548759464057457, 3.335431208620378, 3.800698299305128, 4.408040460557867, 5.346728010207325, 5.637224630071349, 6.137942597437229, 6.769556199789096, 7.401793990800578, 7.689786858108362, 8.171832963629551, 9.057193474583628, 9.379185072505910, 9.797309320659902, 10.34181303107336, 10.67745184781601, 11.59959800722575, 11.80353524638011, 12.03811578497620, 12.57972610085808, 13.33609257513024, 13.84794995635352

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