Properties

Label 2-2400-1.1-c3-0-34
Degree $2$
Conductor $2400$
Sign $1$
Analytic cond. $141.604$
Root an. cond. $11.8997$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·3-s + 4·7-s + 9·9-s + 20·11-s − 70·13-s − 90·17-s + 140·19-s + 12·21-s + 192·23-s + 27·27-s − 134·29-s + 100·31-s + 60·33-s + 170·37-s − 210·39-s − 110·41-s − 532·43-s + 56·47-s − 327·49-s − 270·51-s + 430·53-s + 420·57-s − 20·59-s + 270·61-s + 36·63-s + 524·67-s + 576·69-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.215·7-s + 1/3·9-s + 0.548·11-s − 1.49·13-s − 1.28·17-s + 1.69·19-s + 0.124·21-s + 1.74·23-s + 0.192·27-s − 0.858·29-s + 0.579·31-s + 0.316·33-s + 0.755·37-s − 0.862·39-s − 0.419·41-s − 1.88·43-s + 0.173·47-s − 0.953·49-s − 0.741·51-s + 1.11·53-s + 0.975·57-s − 0.0441·59-s + 0.566·61-s + 0.0719·63-s + 0.955·67-s + 1.00·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2400\)    =    \(2^{5} \cdot 3 \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(141.604\)
Root analytic conductor: \(11.8997\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2400,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(2.824689803\)
\(L(\frac12)\) \(\approx\) \(2.824689803\)
\(L(\frac{5}{2})\) 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 \( 1 - p T \)
5 \( 1 \)
good7 \( 1 - 4 T + p^{3} T^{2} \)
11 \( 1 - 20 T + p^{3} T^{2} \)
13 \( 1 + 70 T + p^{3} T^{2} \)
17 \( 1 + 90 T + p^{3} T^{2} \)
19 \( 1 - 140 T + p^{3} T^{2} \)
23 \( 1 - 192 T + p^{3} T^{2} \)
29 \( 1 + 134 T + p^{3} T^{2} \)
31 \( 1 - 100 T + p^{3} T^{2} \)
37 \( 1 - 170 T + p^{3} T^{2} \)
41 \( 1 + 110 T + p^{3} T^{2} \)
43 \( 1 + 532 T + p^{3} T^{2} \)
47 \( 1 - 56 T + p^{3} T^{2} \)
53 \( 1 - 430 T + p^{3} T^{2} \)
59 \( 1 + 20 T + p^{3} T^{2} \)
61 \( 1 - 270 T + p^{3} T^{2} \)
67 \( 1 - 524 T + p^{3} T^{2} \)
71 \( 1 + 80 T + p^{3} T^{2} \)
73 \( 1 + 330 T + p^{3} T^{2} \)
79 \( 1 - 1060 T + p^{3} T^{2} \)
83 \( 1 - 1188 T + p^{3} T^{2} \)
89 \( 1 - 1274 T + p^{3} T^{2} \)
97 \( 1 - 590 T + p^{3} T^{2} \)
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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

−8.700196882354775773405044746726, −7.78672660643866095425048345119, −7.13171690189338556578977893176, −6.55295262750073746762466313352, −5.16175340142607177249322261449, −4.79241803150896758416144828458, −3.63016708480238565992169708232, −2.79867816986680335918374529496, −1.89730149165633659854667702316, −0.72449200980726547448426695406, 0.72449200980726547448426695406, 1.89730149165633659854667702316, 2.79867816986680335918374529496, 3.63016708480238565992169708232, 4.79241803150896758416144828458, 5.16175340142607177249322261449, 6.55295262750073746762466313352, 7.13171690189338556578977893176, 7.78672660643866095425048345119, 8.700196882354775773405044746726

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