Properties

Label 2-300-1.1-c3-0-0
Degree $2$
Conductor $300$
Sign $1$
Analytic cond. $17.7005$
Root an. cond. $4.20720$
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 − 13·7-s + 9·9-s + 6·11-s + 5·13-s − 78·17-s + 65·19-s + 39·21-s + 138·23-s − 27·27-s + 66·29-s + 299·31-s − 18·33-s − 214·37-s − 15·39-s + 360·41-s + 203·43-s + 78·47-s − 174·49-s + 234·51-s + 636·53-s − 195·57-s + 786·59-s + 467·61-s − 117·63-s − 217·67-s − 414·69-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.701·7-s + 1/3·9-s + 0.164·11-s + 0.106·13-s − 1.11·17-s + 0.784·19-s + 0.405·21-s + 1.25·23-s − 0.192·27-s + 0.422·29-s + 1.73·31-s − 0.0949·33-s − 0.950·37-s − 0.0615·39-s + 1.37·41-s + 0.719·43-s + 0.242·47-s − 0.507·49-s + 0.642·51-s + 1.64·53-s − 0.453·57-s + 1.73·59-s + 0.980·61-s − 0.233·63-s − 0.395·67-s − 0.722·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(17.7005\)
Root analytic conductor: \(4.20720\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(1.324359433\)
\(L(\frac12)\) \(\approx\) \(1.324359433\)
\(L(\frac{5}{2})\) 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 \( 1 + 13 T + p^{3} T^{2} \)
11 \( 1 - 6 T + p^{3} T^{2} \)
13 \( 1 - 5 T + p^{3} T^{2} \)
17 \( 1 + 78 T + p^{3} T^{2} \)
19 \( 1 - 65 T + p^{3} T^{2} \)
23 \( 1 - 6 p T + p^{3} T^{2} \)
29 \( 1 - 66 T + p^{3} T^{2} \)
31 \( 1 - 299 T + p^{3} T^{2} \)
37 \( 1 + 214 T + p^{3} T^{2} \)
41 \( 1 - 360 T + p^{3} T^{2} \)
43 \( 1 - 203 T + p^{3} T^{2} \)
47 \( 1 - 78 T + p^{3} T^{2} \)
53 \( 1 - 12 p T + p^{3} T^{2} \)
59 \( 1 - 786 T + p^{3} T^{2} \)
61 \( 1 - 467 T + p^{3} T^{2} \)
67 \( 1 + 217 T + p^{3} T^{2} \)
71 \( 1 + 360 T + p^{3} T^{2} \)
73 \( 1 + 286 T + p^{3} T^{2} \)
79 \( 1 - 272 T + p^{3} T^{2} \)
83 \( 1 - 6 p T + p^{3} T^{2} \)
89 \( 1 + p^{3} T^{2} \)
97 \( 1 + 511 T + p^{3} T^{2} \)
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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

−11.33084464678479667356745446546, −10.42744002071793728210232350532, −9.500193291056395887631122352923, −8.568511882645077864369180582756, −7.16143638687420049851341504230, −6.45332309417483332528698938193, −5.32132727864424098005152775626, −4.15018764099051957183511425453, −2.72466153288255976674702044996, −0.832408142692456753905514203754, 0.832408142692456753905514203754, 2.72466153288255976674702044996, 4.15018764099051957183511425453, 5.32132727864424098005152775626, 6.45332309417483332528698938193, 7.16143638687420049851341504230, 8.568511882645077864369180582756, 9.500193291056395887631122352923, 10.42744002071793728210232350532, 11.33084464678479667356745446546

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