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Properties

Label 2-72450-1.1-c1-0-49
Degree $2$
Conductor $72450$
Sign $1$
Analytic cond. $578.516$
Root an. cond. $24.0523$
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 + 4-s − 7-s + 8-s + 6·13-s − 14-s + 16-s − 6·17-s + 4·19-s + 23-s + 6·26-s − 28-s + 6·29-s + 32-s − 6·34-s + 10·37-s + 4·38-s − 2·41-s + 8·43-s + 46-s + 8·47-s + 49-s + 6·52-s − 10·53-s − 56-s + 6·58-s + 4·59-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 0.377·7-s + 0.353·8-s + 1.66·13-s − 0.267·14-s + 1/4·16-s − 1.45·17-s + 0.917·19-s + 0.208·23-s + 1.17·26-s − 0.188·28-s + 1.11·29-s + 0.176·32-s − 1.02·34-s + 1.64·37-s + 0.648·38-s − 0.312·41-s + 1.21·43-s + 0.147·46-s + 1.16·47-s + 1/7·49-s + 0.832·52-s − 1.37·53-s − 0.133·56-s + 0.787·58-s + 0.520·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 72450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(72450\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7 \cdot 23\)
Sign: $1$
Analytic conductor: \(578.516\)
Root analytic conductor: \(24.0523\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 72450,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.060451418\)
\(L(\frac12)\) \(\approx\) \(5.060451418\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 \)
7 \( 1 + T \)
23 \( 1 - T \)
good11 \( 1 + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 - 16 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−14.02782971984247, −13.66256758282473, −13.01258152144369, −12.84517262334303, −12.20902272597069, −11.50014648967224, −11.06557076633112, −10.94468649222306, −10.11405328302320, −9.500535350460035, −9.114670997130415, −8.259459301863200, −8.168227356760841, −7.210095144093288, −6.638076479862298, −6.405195841418595, −5.713000102233709, −5.232837044285088, −4.477192085225691, −3.949562229459303, −3.548399302166404, −2.668343097565421, −2.335184500333440, −1.243288555666687, −0.7289615713490493, 0.7289615713490493, 1.243288555666687, 2.335184500333440, 2.668343097565421, 3.548399302166404, 3.949562229459303, 4.477192085225691, 5.232837044285088, 5.713000102233709, 6.405195841418595, 6.638076479862298, 7.210095144093288, 8.168227356760841, 8.259459301863200, 9.114670997130415, 9.500535350460035, 10.11405328302320, 10.94468649222306, 11.06557076633112, 11.50014648967224, 12.20902272597069, 12.84517262334303, 13.01258152144369, 13.66256758282473, 14.02782971984247

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