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Properties

Label 2-46200-1.1-c1-0-50
Degree $2$
Conductor $46200$
Sign $-1$
Analytic cond. $368.908$
Root an. cond. $19.2070$
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  − 3-s + 7-s + 9-s + 11-s − 6·13-s − 6·17-s − 21-s + 8·23-s − 27-s − 2·29-s + 4·31-s − 33-s + 2·37-s + 6·39-s − 2·41-s + 12·43-s − 12·47-s + 49-s + 6·51-s − 6·53-s + 4·59-s + 6·61-s + 63-s − 4·67-s − 8·69-s + 2·73-s + 77-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.377·7-s + 1/3·9-s + 0.301·11-s − 1.66·13-s − 1.45·17-s − 0.218·21-s + 1.66·23-s − 0.192·27-s − 0.371·29-s + 0.718·31-s − 0.174·33-s + 0.328·37-s + 0.960·39-s − 0.312·41-s + 1.82·43-s − 1.75·47-s + 1/7·49-s + 0.840·51-s − 0.824·53-s + 0.520·59-s + 0.768·61-s + 0.125·63-s − 0.488·67-s − 0.963·69-s + 0.234·73-s + 0.113·77-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 46200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(46200\)    =    \(2^{3} \cdot 3 \cdot 5^{2} \cdot 7 \cdot 11\)
Sign: $-1$
Analytic conductor: \(368.908\)
Root analytic conductor: \(19.2070\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 46200,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
5 \( 1 \)
7 \( 1 - T \)
11 \( 1 - T \)
good13 \( 1 + 6 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 18 T + p T^{2} \)
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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

−14.78639943956108, −14.60276586119277, −13.75353831850541, −13.31703526281622, −12.63450283238931, −12.42436595938574, −11.68239287314488, −11.14808962437822, −10.99420005037796, −10.18689799243499, −9.611862946840700, −9.232828348070471, −8.595983014660222, −7.918151473480813, −7.328737503811023, −6.797114278165414, −6.469458739114626, −5.552285192487915, −5.048893973704271, −4.569677701283476, −4.114857094712729, −3.053489129042982, −2.485788545941438, −1.779922609644194, −0.8638352529722764, 0, 0.8638352529722764, 1.779922609644194, 2.485788545941438, 3.053489129042982, 4.114857094712729, 4.569677701283476, 5.048893973704271, 5.552285192487915, 6.469458739114626, 6.797114278165414, 7.328737503811023, 7.918151473480813, 8.595983014660222, 9.232828348070471, 9.611862946840700, 10.18689799243499, 10.99420005037796, 11.14808962437822, 11.68239287314488, 12.42436595938574, 12.63450283238931, 13.31703526281622, 13.75353831850541, 14.60276586119277, 14.78639943956108

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