Properties

Label 2-289800-1.1-c1-0-123
Degree $2$
Conductor $289800$
Sign $-1$
Analytic cond. $2314.06$
Root an. cond. $48.1047$
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  − 7-s + 4·11-s + 2·13-s + 2·17-s − 4·19-s − 23-s + 2·29-s + 8·31-s − 6·37-s + 6·41-s + 4·43-s + 8·47-s + 49-s − 10·53-s + 12·59-s + 14·61-s + 12·67-s − 16·71-s + 14·73-s − 4·77-s + 8·79-s − 12·83-s + 14·89-s − 2·91-s − 2·97-s + 101-s + 103-s + ⋯
L(s)  = 1  − 0.377·7-s + 1.20·11-s + 0.554·13-s + 0.485·17-s − 0.917·19-s − 0.208·23-s + 0.371·29-s + 1.43·31-s − 0.986·37-s + 0.937·41-s + 0.609·43-s + 1.16·47-s + 1/7·49-s − 1.37·53-s + 1.56·59-s + 1.79·61-s + 1.46·67-s − 1.89·71-s + 1.63·73-s − 0.455·77-s + 0.900·79-s − 1.31·83-s + 1.48·89-s − 0.209·91-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 289800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(289800\)    =    \(2^{3} \cdot 3^{2} \cdot 5^{2} \cdot 7 \cdot 23\)
Sign: $-1$
Analytic conductor: \(2314.06\)
Root analytic conductor: \(48.1047\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{289800} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 289800,\ (\ :1/2),\ -1)\)

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 \( 1 \)
7 \( 1 + T \)
23 \( 1 + T \)
good11 \( 1 - 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 16 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−12.91842339943124, −12.45237742115623, −12.03561191253665, −11.64414315117296, −11.14831132724318, −10.58336739234491, −10.27670907880452, −9.630118382824937, −9.337306015571620, −8.774875747777251, −8.315223492132235, −8.002435844116424, −7.230084527137819, −6.709564270139532, −6.480507621376269, −5.929113435338354, −5.452558375055819, −4.778516059772327, −4.179087442193471, −3.806918455860903, −3.394505186508651, −2.501263028556129, −2.243262841263503, −1.163299224748208, −1.020851132250599, 0, 1.020851132250599, 1.163299224748208, 2.243262841263503, 2.501263028556129, 3.394505186508651, 3.806918455860903, 4.179087442193471, 4.778516059772327, 5.452558375055819, 5.929113435338354, 6.480507621376269, 6.709564270139532, 7.230084527137819, 8.002435844116424, 8.315223492132235, 8.774875747777251, 9.337306015571620, 9.630118382824937, 10.27670907880452, 10.58336739234491, 11.14831132724318, 11.64414315117296, 12.03561191253665, 12.45237742115623, 12.91842339943124

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