Properties

Label 2-4800-1.1-c1-0-62
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 3·7-s + 9-s + 2·11-s − 13-s + 2·17-s − 5·19-s − 3·21-s − 6·23-s − 27-s − 10·29-s + 3·31-s − 2·33-s − 2·37-s + 39-s − 8·41-s + 43-s − 2·47-s + 2·49-s − 2·51-s + 4·53-s + 5·57-s − 10·59-s − 7·61-s + 3·63-s − 3·67-s + 6·69-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.13·7-s + 1/3·9-s + 0.603·11-s − 0.277·13-s + 0.485·17-s − 1.14·19-s − 0.654·21-s − 1.25·23-s − 0.192·27-s − 1.85·29-s + 0.538·31-s − 0.348·33-s − 0.328·37-s + 0.160·39-s − 1.24·41-s + 0.152·43-s − 0.291·47-s + 2/7·49-s − 0.280·51-s + 0.549·53-s + 0.662·57-s − 1.30·59-s − 0.896·61-s + 0.377·63-s − 0.366·67-s + 0.722·69-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 4800,\ (\ :1/2),\ -1)\)

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 \( 1 \)
good7 \( 1 - 3 T + p T^{2} \)
11 \( 1 - 2 T + p T^{2} \)
13 \( 1 + T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 5 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 10 T + p T^{2} \)
31 \( 1 - 3 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 8 T + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 - 4 T + p T^{2} \)
59 \( 1 + 10 T + p T^{2} \)
61 \( 1 + 7 T + p T^{2} \)
67 \( 1 + 3 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 - 17 T + p T^{2} \)
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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

−7.87914659835863207294698651709, −7.26683856545720539350458472989, −6.35078788674859422957686456838, −5.76958242735491544916508279271, −4.92993401446228872799172657878, −4.30861693097863502278050736122, −3.51380796888682305453962490670, −2.09699607926453765135786428693, −1.48244217054512001330990913109, 0, 1.48244217054512001330990913109, 2.09699607926453765135786428693, 3.51380796888682305453962490670, 4.30861693097863502278050736122, 4.92993401446228872799172657878, 5.76958242735491544916508279271, 6.35078788674859422957686456838, 7.26683856545720539350458472989, 7.87914659835863207294698651709

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