Properties

Label 2-46800-1.1-c1-0-74
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 + 2·11-s + 13-s − 4·17-s − 2·19-s − 6·23-s + 2·29-s + 4·31-s − 6·37-s + 6·41-s − 8·43-s + 8·47-s + 9·49-s − 10·53-s − 14·59-s + 10·61-s − 4·67-s + 8·71-s + 10·73-s − 8·77-s + 8·79-s + 12·83-s + 18·89-s − 4·91-s + 6·97-s + 101-s + 103-s + ⋯
L(s)  = 1  − 1.51·7-s + 0.603·11-s + 0.277·13-s − 0.970·17-s − 0.458·19-s − 1.25·23-s + 0.371·29-s + 0.718·31-s − 0.986·37-s + 0.937·41-s − 1.21·43-s + 1.16·47-s + 9/7·49-s − 1.37·53-s − 1.82·59-s + 1.28·61-s − 0.488·67-s + 0.949·71-s + 1.17·73-s − 0.911·77-s + 0.900·79-s + 1.31·83-s + 1.90·89-s − 0.419·91-s + 0.609·97-s + 0.0995·101-s + 0.0985·103-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 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 + 14 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 18 T + p T^{2} \)
97 \( 1 - 6 T + p T^{2} \)
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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

−14.99506682639868, −14.11460917146439, −13.85637600157883, −13.34713088148632, −12.76632888491827, −12.30344947994079, −11.93402680778117, −11.20092852290190, −10.63192136292347, −10.17050527016246, −9.584999116055300, −9.164447824314577, −8.659398319773836, −8.021422326448310, −7.371127574780059, −6.542661427260019, −6.411696422030291, −5.990756565334416, −5.006310075487603, −4.426923758225721, −3.665757901087476, −3.391015810086760, −2.449876756732225, −1.917849228666232, −0.8013509446578811, 0, 0.8013509446578811, 1.917849228666232, 2.449876756732225, 3.391015810086760, 3.665757901087476, 4.426923758225721, 5.006310075487603, 5.990756565334416, 6.411696422030291, 6.542661427260019, 7.371127574780059, 8.021422326448310, 8.659398319773836, 9.164447824314577, 9.584999116055300, 10.17050527016246, 10.63192136292347, 11.20092852290190, 11.93402680778117, 12.30344947994079, 12.76632888491827, 13.34713088148632, 13.85637600157883, 14.11460917146439, 14.99506682639868

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