Properties

Label 2-46800-1.1-c1-0-49
Degree $2$
Conductor $46800$
Sign $-1$
Analytic cond. $373.699$
Root an. cond. $19.3313$
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  − 4·7-s − 6·11-s − 13-s − 6·17-s − 2·19-s − 6·23-s + 6·29-s − 2·31-s − 2·37-s + 6·41-s + 2·43-s + 12·47-s + 9·49-s + 6·53-s + 6·59-s + 2·61-s − 4·67-s − 6·71-s + 10·73-s + 24·77-s + 4·79-s + 6·89-s + 4·91-s − 2·97-s + 101-s + 103-s + 107-s + ⋯
L(s)  = 1  − 1.51·7-s − 1.80·11-s − 0.277·13-s − 1.45·17-s − 0.458·19-s − 1.25·23-s + 1.11·29-s − 0.359·31-s − 0.328·37-s + 0.937·41-s + 0.304·43-s + 1.75·47-s + 9/7·49-s + 0.824·53-s + 0.781·59-s + 0.256·61-s − 0.488·67-s − 0.712·71-s + 1.17·73-s + 2.73·77-s + 0.450·79-s + 0.635·89-s + 0.419·91-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 46800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(46800\)    =    \(2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 13\)
Sign: $-1$
Analytic conductor: \(373.699\)
Root analytic conductor: \(19.3313\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{46800} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 46800,\ (\ :1/2),\ -1)\)

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 \( 1 \)
5 \( 1 \)
13 \( 1 + T \)
good7 \( 1 + 4 T + p T^{2} \)
11 \( 1 + 6 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−15.05451327272720, −14.22223558021067, −13.67169599848707, −13.19335155124104, −13.00044606640348, −12.19149328186992, −12.11189493797527, −11.00234998660564, −10.61382332862049, −10.27795267946377, −9.669665446267767, −9.151077117585799, −8.561353640336940, −7.987447741768773, −7.406191046616001, −6.761948628162011, −6.370542725013739, −5.657029836741539, −5.249101398558105, −4.271083913236459, −4.003007462446318, −2.973278467413552, −2.546346568278101, −2.097114405507648, −0.6244316626413631, 0, 0.6244316626413631, 2.097114405507648, 2.546346568278101, 2.973278467413552, 4.003007462446318, 4.271083913236459, 5.249101398558105, 5.657029836741539, 6.370542725013739, 6.761948628162011, 7.406191046616001, 7.987447741768773, 8.561353640336940, 9.151077117585799, 9.669665446267767, 10.27795267946377, 10.61382332862049, 11.00234998660564, 12.11189493797527, 12.19149328186992, 13.00044606640348, 13.19335155124104, 13.67169599848707, 14.22223558021067, 15.05451327272720

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