Properties

Label 2-46800-1.1-c1-0-10
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 7-s − 5·11-s − 13-s − 5·17-s + 7·29-s + 9·31-s − 8·37-s + 2·41-s − 8·43-s − 9·47-s − 6·49-s − 11·53-s + 59-s − 7·61-s + 15·67-s − 8·71-s + 4·73-s − 5·77-s + 4·79-s − 9·83-s − 16·89-s − 91-s + 2·97-s + 101-s + 103-s + 107-s + 109-s + ⋯
L(s)  = 1  + 0.377·7-s − 1.50·11-s − 0.277·13-s − 1.21·17-s + 1.29·29-s + 1.61·31-s − 1.31·37-s + 0.312·41-s − 1.21·43-s − 1.31·47-s − 6/7·49-s − 1.51·53-s + 0.130·59-s − 0.896·61-s + 1.83·67-s − 0.949·71-s + 0.468·73-s − 0.569·77-s + 0.450·79-s − 0.987·83-s − 1.69·89-s − 0.104·91-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-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: \(0\)
Selberg data: \((2,\ 46800,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.013178422\)
\(L(\frac12)\) \(\approx\) \(1.013178422\)
\(L(\frac{3}{2})\) 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 \( 1 - T + p T^{2} \)
11 \( 1 + 5 T + p T^{2} \)
17 \( 1 + 5 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 7 T + p T^{2} \)
31 \( 1 - 9 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + 9 T + p T^{2} \)
53 \( 1 + 11 T + p T^{2} \)
59 \( 1 - T + p T^{2} \)
61 \( 1 + 7 T + p T^{2} \)
67 \( 1 - 15 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + 9 T + p T^{2} \)
89 \( 1 + 16 T + p T^{2} \)
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

−14.53922737976128, −14.00672302674831, −13.63019245676803, −13.00354079767420, −12.67775592229813, −11.99960200701639, −11.45626606789538, −10.98423787626966, −10.40866891938719, −10.00514710216726, −9.471910574532064, −8.610938203325081, −8.189011241200523, −7.973274688615076, −7.013727857193436, −6.666254717659236, −6.027233105378678, −5.195573364409513, −4.770647509746203, −4.454116471591201, −3.325305461406003, −2.855459461487458, −2.183846821924556, −1.475009960691060, −0.3465148229577240, 0.3465148229577240, 1.475009960691060, 2.183846821924556, 2.855459461487458, 3.325305461406003, 4.454116471591201, 4.770647509746203, 5.195573364409513, 6.027233105378678, 6.666254717659236, 7.013727857193436, 7.973274688615076, 8.189011241200523, 8.610938203325081, 9.471910574532064, 10.00514710216726, 10.40866891938719, 10.98423787626966, 11.45626606789538, 11.99960200701639, 12.67775592229813, 13.00354079767420, 13.63019245676803, 14.00672302674831, 14.53922737976128

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