Properties

Label 2-46800-1.1-c1-0-4
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  − 5·7-s + 11-s − 13-s − 3·17-s − 6·19-s + 7·23-s − 6·29-s + 2·31-s − 37-s − 7·41-s + 8·43-s − 2·47-s + 18·49-s + 13·53-s − 8·59-s − 7·61-s + 12·67-s + 71-s + 10·73-s − 5·77-s − 3·79-s − 8·83-s − 13·89-s + 5·91-s − 7·97-s + 101-s + 103-s + ⋯
L(s)  = 1  − 1.88·7-s + 0.301·11-s − 0.277·13-s − 0.727·17-s − 1.37·19-s + 1.45·23-s − 1.11·29-s + 0.359·31-s − 0.164·37-s − 1.09·41-s + 1.21·43-s − 0.291·47-s + 18/7·49-s + 1.78·53-s − 1.04·59-s − 0.896·61-s + 1.46·67-s + 0.118·71-s + 1.17·73-s − 0.569·77-s − 0.337·79-s − 0.878·83-s − 1.37·89-s + 0.524·91-s − 0.710·97-s + 0.0995·101-s + 0.0985·103-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

\(L(1)\) \(\approx\) \(0.5719051534\)
\(L(\frac12)\) \(\approx\) \(0.5719051534\)
\(L(\frac{3}{2})\) 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 \( 1 + 5 T + p T^{2} \)
11 \( 1 - T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 - 7 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 + T + p T^{2} \)
41 \( 1 + 7 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 - 13 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 + 7 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 3 T + p T^{2} \)
83 \( 1 + 8 T + p T^{2} \)
89 \( 1 + 13 T + p T^{2} \)
97 \( 1 + 7 T + p T^{2} \)
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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

−14.90975553456112, −13.88199499078302, −13.49321092390972, −13.04877748142736, −12.50247456353213, −12.34909410585243, −11.43017528084470, −10.88021747133657, −10.50721873918392, −9.715072487211490, −9.501868934637533, −8.794549856206188, −8.557674186884738, −7.524218851928767, −6.955289657149566, −6.644296641371498, −6.119121696744682, −5.485120277656157, −4.758684978345879, −3.938156379837562, −3.656690887945405, −2.712548668862920, −2.429350434669438, −1.315192294027169, −0.2718590986394657, 0.2718590986394657, 1.315192294027169, 2.429350434669438, 2.712548668862920, 3.656690887945405, 3.938156379837562, 4.758684978345879, 5.485120277656157, 6.119121696744682, 6.644296641371498, 6.955289657149566, 7.524218851928767, 8.557674186884738, 8.794549856206188, 9.501868934637533, 9.715072487211490, 10.50721873918392, 10.88021747133657, 11.43017528084470, 12.34909410585243, 12.50247456353213, 13.04877748142736, 13.49321092390972, 13.88199499078302, 14.90975553456112

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