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Properties

Label 2-44400-1.1-c1-0-44
Degree $2$
Conductor $44400$
Sign $-1$
Analytic cond. $354.535$
Root an. cond. $18.8291$
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  + 3-s − 7-s + 9-s − 3·11-s − 2·13-s − 3·17-s − 2·19-s − 21-s + 27-s − 3·29-s + 31-s − 3·33-s − 37-s − 2·39-s + 9·41-s + 11·43-s − 6·49-s − 3·51-s + 9·53-s − 2·57-s + 6·59-s − 61-s − 63-s + 8·67-s + 12·71-s − 8·73-s + 3·77-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.377·7-s + 1/3·9-s − 0.904·11-s − 0.554·13-s − 0.727·17-s − 0.458·19-s − 0.218·21-s + 0.192·27-s − 0.557·29-s + 0.179·31-s − 0.522·33-s − 0.164·37-s − 0.320·39-s + 1.40·41-s + 1.67·43-s − 6/7·49-s − 0.420·51-s + 1.23·53-s − 0.264·57-s + 0.781·59-s − 0.128·61-s − 0.125·63-s + 0.977·67-s + 1.42·71-s − 0.936·73-s + 0.341·77-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 44400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 44400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(44400\)    =    \(2^{4} \cdot 3 \cdot 5^{2} \cdot 37\)
Sign: $-1$
Analytic conductor: \(354.535\)
Root analytic conductor: \(18.8291\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 44400,\ (\ :1/2),\ -1)\)

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 \( 1 \)
3 \( 1 - T \)
5 \( 1 \)
37 \( 1 + T \)
good7 \( 1 + T + p T^{2} \)
11 \( 1 + 3 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 - T + p T^{2} \)
41 \( 1 - 9 T + p T^{2} \)
43 \( 1 - 11 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 + T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 8 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 13 T + p T^{2} \)
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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

−14.78145619199207, −14.52608732794431, −13.86508302397453, −13.31034475639095, −12.90470972947814, −12.53183305251848, −11.91208858983925, −11.09764967694815, −10.80936754824640, −10.16849998930356, −9.594747880697294, −9.213772030620743, −8.555950996512597, −8.049564418740376, −7.480029903216625, −7.003227693577329, −6.372479470477543, −5.668349856862384, −5.139757085571614, −4.335428052358377, −3.950745309927973, −3.074665644442312, −2.447036930373427, −2.104734091321414, −0.9199351799255645, 0, 0.9199351799255645, 2.104734091321414, 2.447036930373427, 3.074665644442312, 3.950745309927973, 4.335428052358377, 5.139757085571614, 5.668349856862384, 6.372479470477543, 7.003227693577329, 7.480029903216625, 8.049564418740376, 8.555950996512597, 9.213772030620743, 9.594747880697294, 10.16849998930356, 10.80936754824640, 11.09764967694815, 11.91208858983925, 12.53183305251848, 12.90470972947814, 13.31034475639095, 13.86508302397453, 14.52608732794431, 14.78145619199207

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