Properties

Label 2-202800-1.1-c1-0-115
Degree $2$
Conductor $202800$
Sign $1$
Analytic cond. $1619.36$
Root an. cond. $40.2413$
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  + 3-s + 2·7-s + 9-s + 4·17-s + 6·19-s + 2·21-s + 6·23-s + 27-s + 4·29-s + 8·31-s − 6·37-s − 6·41-s + 4·43-s − 8·47-s − 3·49-s + 4·51-s − 2·53-s + 6·57-s − 2·61-s + 2·63-s + 4·67-s + 6·69-s − 8·71-s − 16·79-s + 81-s + 4·83-s + 4·87-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.755·7-s + 1/3·9-s + 0.970·17-s + 1.37·19-s + 0.436·21-s + 1.25·23-s + 0.192·27-s + 0.742·29-s + 1.43·31-s − 0.986·37-s − 0.937·41-s + 0.609·43-s − 1.16·47-s − 3/7·49-s + 0.560·51-s − 0.274·53-s + 0.794·57-s − 0.256·61-s + 0.251·63-s + 0.488·67-s + 0.722·69-s − 0.949·71-s − 1.80·79-s + 1/9·81-s + 0.439·83-s + 0.428·87-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 202800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 202800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(202800\)    =    \(2^{4} \cdot 3 \cdot 5^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(1619.36\)
Root analytic conductor: \(40.2413\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{202800} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 202800,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.042154335\)
\(L(\frac12)\) \(\approx\) \(5.042154335\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - T \)
5 \( 1 \)
13 \( 1 \)
good7 \( 1 - 2 T + p T^{2} \)
11 \( 1 + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 4 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
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

−13.17019131235612, −12.56019925443132, −12.03957651872658, −11.70174133649414, −11.28279649124292, −10.63490227296293, −10.11808089656180, −9.850410024006262, −9.218985687240331, −8.751805258850048, −8.261501683892336, −7.869661162167494, −7.410372573138330, −6.894235430776494, −6.395726636540060, −5.673821057646595, −5.062232944291214, −4.871015558475023, −4.211233221965633, −3.389753520336723, −3.107773798382557, −2.607427480174917, −1.645095396894785, −1.315207677091654, −0.6419727143190675, 0.6419727143190675, 1.315207677091654, 1.645095396894785, 2.607427480174917, 3.107773798382557, 3.389753520336723, 4.211233221965633, 4.871015558475023, 5.062232944291214, 5.673821057646595, 6.395726636540060, 6.894235430776494, 7.410372573138330, 7.869661162167494, 8.261501683892336, 8.751805258850048, 9.218985687240331, 9.850410024006262, 10.11808089656180, 10.63490227296293, 11.28279649124292, 11.70174133649414, 12.03957651872658, 12.56019925443132, 13.17019131235612

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