Properties

Label 2-600-1.1-c3-0-11
Degree $2$
Conductor $600$
Sign $1$
Analytic cond. $35.4011$
Root an. cond. $5.94988$
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 + 5·7-s + 9·9-s + 14·11-s + 13-s + 46·17-s + 19·19-s + 15·21-s − 46·23-s + 27·27-s + 14·29-s + 133·31-s + 42·33-s + 258·37-s + 3·39-s + 84·41-s − 167·43-s + 410·47-s − 318·49-s + 138·51-s + 456·53-s + 57·57-s − 194·59-s − 17·61-s + 45·63-s + 653·67-s − 138·69-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.269·7-s + 1/3·9-s + 0.383·11-s + 0.0213·13-s + 0.656·17-s + 0.229·19-s + 0.155·21-s − 0.417·23-s + 0.192·27-s + 0.0896·29-s + 0.770·31-s + 0.221·33-s + 1.14·37-s + 0.0123·39-s + 0.319·41-s − 0.592·43-s + 1.27·47-s − 0.927·49-s + 0.378·51-s + 1.18·53-s + 0.132·57-s − 0.428·59-s − 0.0356·61-s + 0.0899·63-s + 1.19·67-s − 0.240·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(600\)    =    \(2^{3} \cdot 3 \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(35.4011\)
Root analytic conductor: \(5.94988\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 600,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(2.740482475\)
\(L(\frac12)\) \(\approx\) \(2.740482475\)
\(L(\frac{5}{2})\) 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 \( 1 - 5 T + p^{3} T^{2} \)
11 \( 1 - 14 T + p^{3} T^{2} \)
13 \( 1 - T + p^{3} T^{2} \)
17 \( 1 - 46 T + p^{3} T^{2} \)
19 \( 1 - p T + p^{3} T^{2} \)
23 \( 1 + 2 p T + p^{3} T^{2} \)
29 \( 1 - 14 T + p^{3} T^{2} \)
31 \( 1 - 133 T + p^{3} T^{2} \)
37 \( 1 - 258 T + p^{3} T^{2} \)
41 \( 1 - 84 T + p^{3} T^{2} \)
43 \( 1 + 167 T + p^{3} T^{2} \)
47 \( 1 - 410 T + p^{3} T^{2} \)
53 \( 1 - 456 T + p^{3} T^{2} \)
59 \( 1 + 194 T + p^{3} T^{2} \)
61 \( 1 + 17 T + p^{3} T^{2} \)
67 \( 1 - 653 T + p^{3} T^{2} \)
71 \( 1 - 828 T + p^{3} T^{2} \)
73 \( 1 - 570 T + p^{3} T^{2} \)
79 \( 1 + 552 T + p^{3} T^{2} \)
83 \( 1 - 142 T + p^{3} T^{2} \)
89 \( 1 + 1104 T + p^{3} T^{2} \)
97 \( 1 - 841 T + p^{3} T^{2} \)
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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

−10.08739957229836474837263440771, −9.432632991089721204452018249294, −8.429155463864919476718953001235, −7.75940072420533081688199362022, −6.75825932127965148456750396998, −5.69519306085952846566476914670, −4.53211551748147768776432429537, −3.52904289823064767762286064968, −2.35146608250949449895435806483, −1.01640433081996302184915470479, 1.01640433081996302184915470479, 2.35146608250949449895435806483, 3.52904289823064767762286064968, 4.53211551748147768776432429537, 5.69519306085952846566476914670, 6.75825932127965148456750396998, 7.75940072420533081688199362022, 8.429155463864919476718953001235, 9.432632991089721204452018249294, 10.08739957229836474837263440771

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