Properties

Label 2-46800-1.1-c1-0-67
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·7-s − 6·11-s − 13-s + 6·17-s + 4·19-s − 3·23-s + 3·29-s + 4·31-s − 2·37-s − 6·41-s − 7·43-s + 9·49-s − 9·53-s − 6·59-s − 61-s + 14·67-s − 6·71-s + 4·73-s + 24·77-s − 11·79-s + 6·83-s + 4·91-s + 10·97-s + 101-s + 103-s + 107-s + 109-s + ⋯
L(s)  = 1  − 1.51·7-s − 1.80·11-s − 0.277·13-s + 1.45·17-s + 0.917·19-s − 0.625·23-s + 0.557·29-s + 0.718·31-s − 0.328·37-s − 0.937·41-s − 1.06·43-s + 9/7·49-s − 1.23·53-s − 0.781·59-s − 0.128·61-s + 1.71·67-s − 0.712·71-s + 0.468·73-s + 2.73·77-s − 1.23·79-s + 0.658·83-s + 0.419·91-s + 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-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: \(1\)
Selberg data: \((2,\ 46800,\ (\ :1/2),\ -1)\)

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 \( 1 \)
5 \( 1 \)
13 \( 1 + T \)
good7 \( 1 + 4 T + p T^{2} \)
11 \( 1 + 6 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 7 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 + T + p T^{2} \)
67 \( 1 - 14 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 + 11 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 + p T^{2} \)
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

−14.94383714127547, −14.14201369228056, −13.85361975425680, −13.23776764671028, −12.80830019316348, −12.38728094504100, −11.88614241788527, −11.30240468992635, −10.32930477571877, −10.16197108398616, −9.888643965776918, −9.225617982956614, −8.471173933199031, −7.815643912774406, −7.616849108512087, −6.787321282663176, −6.306069851704356, −5.623253013934475, −5.198526825782010, −4.607641188021187, −3.518482932132103, −3.185674535941393, −2.743591114664634, −1.845945081518659, −0.7545717209561546, 0, 0.7545717209561546, 1.845945081518659, 2.743591114664634, 3.185674535941393, 3.518482932132103, 4.607641188021187, 5.198526825782010, 5.623253013934475, 6.306069851704356, 6.787321282663176, 7.616849108512087, 7.815643912774406, 8.471173933199031, 9.225617982956614, 9.888643965776918, 10.16197108398616, 10.32930477571877, 11.30240468992635, 11.88614241788527, 12.38728094504100, 12.80830019316348, 13.23776764671028, 13.85361975425680, 14.14201369228056, 14.94383714127547

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