Properties

Label 2-46800-1.1-c1-0-1
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 − 2·11-s + 13-s + 2·17-s − 6·19-s − 6·23-s − 2·29-s + 6·31-s + 2·37-s − 10·41-s − 10·43-s + 12·47-s + 9·49-s + 2·53-s + 10·59-s + 2·61-s − 12·67-s + 10·71-s − 10·73-s + 8·77-s + 4·79-s + 14·89-s − 4·91-s − 14·97-s + 101-s + 103-s + 107-s + ⋯
L(s)  = 1  − 1.51·7-s − 0.603·11-s + 0.277·13-s + 0.485·17-s − 1.37·19-s − 1.25·23-s − 0.371·29-s + 1.07·31-s + 0.328·37-s − 1.56·41-s − 1.52·43-s + 1.75·47-s + 9/7·49-s + 0.274·53-s + 1.30·59-s + 0.256·61-s − 1.46·67-s + 1.18·71-s − 1.17·73-s + 0.911·77-s + 0.450·79-s + 1.48·89-s − 0.419·91-s − 1.42·97-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

\(L(1)\) \(\approx\) \(0.4578111355\)
\(L(\frac12)\) \(\approx\) \(0.4578111355\)
\(L(\frac{3}{2})\) 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 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 6 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 - 10 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 10 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−14.81468137510506, −13.84422532832874, −13.49911504187212, −13.17111486594036, −12.53501841981274, −12.09292080707274, −11.67983265462451, −10.77694230849397, −10.29262574499205, −10.05957953497791, −9.456081427576616, −8.801406298623342, −8.256963293522670, −7.832792286795169, −6.886417131769271, −6.653867720155729, −5.997655709213787, −5.557894498911867, −4.777021264043756, −3.924330981962506, −3.660601912669189, −2.760445854345499, −2.348615953540351, −1.362341114539833, −0.2385312691037813, 0.2385312691037813, 1.362341114539833, 2.348615953540351, 2.760445854345499, 3.660601912669189, 3.924330981962506, 4.777021264043756, 5.557894498911867, 5.997655709213787, 6.653867720155729, 6.886417131769271, 7.832792286795169, 8.256963293522670, 8.801406298623342, 9.456081427576616, 10.05957953497791, 10.29262574499205, 10.77694230849397, 11.67983265462451, 12.09292080707274, 12.53501841981274, 13.17111486594036, 13.49911504187212, 13.84422532832874, 14.81468137510506

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