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Properties

Label 2-46800-1.1-c1-0-100
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  + 13-s + 2·17-s − 4·23-s − 6·29-s − 8·31-s − 10·37-s + 6·41-s + 4·43-s + 8·47-s − 7·49-s + 2·53-s + 6·61-s + 12·67-s + 8·71-s − 6·73-s − 8·79-s + 12·83-s + 14·89-s + 10·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  + 0.277·13-s + 0.485·17-s − 0.834·23-s − 1.11·29-s − 1.43·31-s − 1.64·37-s + 0.937·41-s + 0.609·43-s + 1.16·47-s − 49-s + 0.274·53-s + 0.768·61-s + 1.46·67-s + 0.949·71-s − 0.702·73-s − 0.900·79-s + 1.31·83-s + 1.48·89-s + 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-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: Trivial
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 \( 1 + p T^{2} \)
11 \( 1 + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
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 \( 1 - 6 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−14.65487432362289, −14.44812774884020, −13.88184379485391, −13.29208947399891, −12.76949271601307, −12.34333236706794, −11.77930190796713, −11.21002527579151, −10.69259341866729, −10.28757586748985, −9.461223787511946, −9.256548574629677, −8.517213267438565, −7.988370637192036, −7.393035070227441, −6.972709051890204, −6.196474040236003, −5.626818864784440, −5.257893006721652, −4.412017844891959, −3.671929736613518, −3.450859240783671, −2.328273429950216, −1.885065735850373, −0.9531481514451990, 0, 0.9531481514451990, 1.885065735850373, 2.328273429950216, 3.450859240783671, 3.671929736613518, 4.412017844891959, 5.257893006721652, 5.626818864784440, 6.196474040236003, 6.972709051890204, 7.393035070227441, 7.988370637192036, 8.517213267438565, 9.256548574629677, 9.461223787511946, 10.28757586748985, 10.69259341866729, 11.21002527579151, 11.77930190796713, 12.34333236706794, 12.76949271601307, 13.29208947399891, 13.88184379485391, 14.44812774884020, 14.65487432362289

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