Properties

Label 2-46800-1.1-c1-0-8
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  − 4·7-s + 11-s + 13-s − 7·17-s + 3·19-s + 4·29-s − 6·31-s + 8·37-s + 5·41-s − 4·43-s − 12·47-s + 9·49-s − 10·53-s + 4·59-s + 8·61-s − 9·67-s − 8·71-s − 13·73-s − 4·77-s − 8·79-s − 3·83-s + 11·89-s − 4·91-s + 10·97-s + 101-s + 103-s + 107-s + ⋯
L(s)  = 1  − 1.51·7-s + 0.301·11-s + 0.277·13-s − 1.69·17-s + 0.688·19-s + 0.742·29-s − 1.07·31-s + 1.31·37-s + 0.780·41-s − 0.609·43-s − 1.75·47-s + 9/7·49-s − 1.37·53-s + 0.520·59-s + 1.02·61-s − 1.09·67-s − 0.949·71-s − 1.52·73-s − 0.455·77-s − 0.900·79-s − 0.329·83-s + 1.16·89-s − 0.419·91-s + 1.01·97-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: \(0\)
Selberg data: \((2,\ 46800,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8811398515\)
\(L(\frac12)\) \(\approx\) \(0.8811398515\)
\(L(\frac{3}{2})\) 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 \( 1 - T + p T^{2} \)
17 \( 1 + 7 T + p T^{2} \)
19 \( 1 - 3 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 4 T + p T^{2} \)
31 \( 1 + 6 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 - 5 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 + 9 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 13 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 3 T + p T^{2} \)
89 \( 1 - 11 T + p T^{2} \)
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

−14.69265852853776, −14.04718325638401, −13.31283864505791, −13.10221620603634, −12.80380443062726, −11.97193323832595, −11.50567356357692, −11.03147461811730, −10.37802562406504, −9.837021389823479, −9.327851728977370, −9.008825704613243, −8.340501893372112, −7.666449424852500, −6.911636490554824, −6.643948593277134, −6.072082832643742, −5.558493197678717, −4.583747133212224, −4.244947267879478, −3.340641847997155, −3.016975158337125, −2.219652613316736, −1.360549925640037, −0.3359802759252648, 0.3359802759252648, 1.360549925640037, 2.219652613316736, 3.016975158337125, 3.340641847997155, 4.244947267879478, 4.583747133212224, 5.558493197678717, 6.072082832643742, 6.643948593277134, 6.911636490554824, 7.666449424852500, 8.340501893372112, 9.008825704613243, 9.327851728977370, 9.837021389823479, 10.37802562406504, 11.03147461811730, 11.50567356357692, 11.97193323832595, 12.80380443062726, 13.10221620603634, 13.31283864505791, 14.04718325638401, 14.69265852853776

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