Properties

Label 2-4800-1.1-c1-0-26
Degree $2$
Conductor $4800$
Sign $1$
Analytic cond. $38.3281$
Root an. cond. $6.19097$
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  − 3-s + 9-s + 4·11-s + 2·13-s + 2·17-s + 8·19-s + 4·23-s − 27-s + 6·29-s − 4·33-s + 2·37-s − 2·39-s − 6·41-s − 4·43-s − 12·47-s − 7·49-s − 2·51-s − 6·53-s − 8·57-s + 12·59-s − 14·61-s + 12·67-s − 4·69-s − 2·73-s + 8·79-s + 81-s + 4·83-s + ⋯
L(s)  = 1  − 0.577·3-s + 1/3·9-s + 1.20·11-s + 0.554·13-s + 0.485·17-s + 1.83·19-s + 0.834·23-s − 0.192·27-s + 1.11·29-s − 0.696·33-s + 0.328·37-s − 0.320·39-s − 0.937·41-s − 0.609·43-s − 1.75·47-s − 49-s − 0.280·51-s − 0.824·53-s − 1.05·57-s + 1.56·59-s − 1.79·61-s + 1.46·67-s − 0.481·69-s − 0.234·73-s + 0.900·79-s + 1/9·81-s + 0.439·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4800\)    =    \(2^{6} \cdot 3 \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(38.3281\)
Root analytic conductor: \(6.19097\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{4800} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4800,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.011033602\)
\(L(\frac12)\) \(\approx\) \(2.011033602\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good7 \( 1 + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−8.237352643961023408661984767701, −7.51093282875750582307294115689, −6.61932386942845090412720839194, −6.30913232533477921240912249858, −5.22691007888024494413904203828, −4.79662531372385296267499567399, −3.63142761694635761529529687946, −3.13129945114593650233364697437, −1.59038547982397983856405985990, −0.894851056729039479934637638740, 0.894851056729039479934637638740, 1.59038547982397983856405985990, 3.13129945114593650233364697437, 3.63142761694635761529529687946, 4.79662531372385296267499567399, 5.22691007888024494413904203828, 6.30913232533477921240912249858, 6.61932386942845090412720839194, 7.51093282875750582307294115689, 8.237352643961023408661984767701

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