Properties

Label 2-46800-1.1-c1-0-43
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  − 2·7-s + 6·11-s + 13-s + 19-s + 6·23-s + 3·29-s − 8·31-s − 37-s + 9·41-s − 8·43-s − 3·47-s − 3·49-s − 3·53-s − 6·59-s − 10·61-s + 13·67-s + 9·71-s − 4·73-s − 12·77-s − 11·79-s + 12·83-s + 6·89-s − 2·91-s − 10·97-s + 101-s + 103-s + 107-s + ⋯
L(s)  = 1  − 0.755·7-s + 1.80·11-s + 0.277·13-s + 0.229·19-s + 1.25·23-s + 0.557·29-s − 1.43·31-s − 0.164·37-s + 1.40·41-s − 1.21·43-s − 0.437·47-s − 3/7·49-s − 0.412·53-s − 0.781·59-s − 1.28·61-s + 1.58·67-s + 1.06·71-s − 0.468·73-s − 1.36·77-s − 1.23·79-s + 1.31·83-s + 0.635·89-s − 0.209·91-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

\(L(1)\) \(\approx\) \(2.491417035\)
\(L(\frac12)\) \(\approx\) \(2.491417035\)
\(L(\frac{3}{2})\) 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 \( 1 + 2 T + p T^{2} \)
11 \( 1 - 6 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + T + p T^{2} \)
41 \( 1 - 9 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + 3 T + p T^{2} \)
53 \( 1 + 3 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 13 T + p T^{2} \)
71 \( 1 - 9 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 + 11 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
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

−14.59502490147825, −14.11560400193585, −13.63925474783846, −12.91033318721297, −12.64437396906349, −12.06114777710786, −11.42225211171160, −11.10313250203081, −10.48846206929434, −9.668600638458168, −9.405734034512672, −8.953642490384527, −8.408091525955990, −7.602171245339063, −7.028093816767445, −6.528748981986207, −6.179101917443573, −5.431455481521163, −4.717119098245104, −4.115190560565334, −3.344485304426403, −3.180004086620502, −2.023903950083853, −1.355613004464540, −0.5977354772602166, 0.5977354772602166, 1.355613004464540, 2.023903950083853, 3.180004086620502, 3.344485304426403, 4.115190560565334, 4.717119098245104, 5.431455481521163, 6.179101917443573, 6.528748981986207, 7.028093816767445, 7.602171245339063, 8.408091525955990, 8.953642490384527, 9.405734034512672, 9.668600638458168, 10.48846206929434, 11.10313250203081, 11.42225211171160, 12.06114777710786, 12.64437396906349, 12.91033318721297, 13.63925474783846, 14.11560400193585, 14.59502490147825

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