Properties

Label 2-300-3.2-c8-0-31
Degree $2$
Conductor $300$
Sign $1$
Analytic cond. $122.213$
Root an. cond. $11.0550$
Motivic weight $8$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 81·3-s + 239·7-s + 6.56e3·9-s − 2.06e4·13-s + 1.00e5·19-s + 1.93e4·21-s + 5.31e5·27-s + 5.83e5·31-s + 5.03e5·37-s − 1.67e6·39-s + 3.34e6·43-s − 5.70e6·49-s + 8.14e6·57-s + 2.41e7·61-s + 1.56e6·63-s + 3.72e7·67-s + 1.61e7·73-s − 1.88e7·79-s + 4.30e7·81-s − 4.93e6·91-s + 4.72e7·93-s − 9.47e7·97-s + 4.44e7·103-s + 6.81e7·109-s + 4.07e7·111-s − 1.35e8·117-s + ⋯
L(s)  = 1  + 3-s + 0.0995·7-s + 9-s − 0.722·13-s + 0.771·19-s + 0.0995·21-s + 27-s + 0.631·31-s + 0.268·37-s − 0.722·39-s + 0.978·43-s − 0.990·49-s + 0.771·57-s + 1.74·61-s + 0.0995·63-s + 1.85·67-s + 0.569·73-s − 0.484·79-s + 81-s − 0.0719·91-s + 0.631·93-s − 1.07·97-s + 0.394·103-s + 0.482·109-s + 0.268·111-s − 0.722·117-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(122.213\)
Root analytic conductor: \(11.0550\)
Motivic weight: \(8\)
Rational: yes
Arithmetic: yes
Character: $\chi_{300} (101, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :4),\ 1)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(3.560210414\)
\(L(\frac12)\) \(\approx\) \(3.560210414\)
\(L(5)\) 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 \( 1 - p^{4} T \)
5 \( 1 \)
good7 \( 1 - 239 T + p^{8} T^{2} \)
11 \( ( 1 - p^{4} T )( 1 + p^{4} T ) \)
13 \( 1 + 20641 T + p^{8} T^{2} \)
17 \( ( 1 - p^{4} T )( 1 + p^{4} T ) \)
19 \( 1 - 100559 T + p^{8} T^{2} \)
23 \( ( 1 - p^{4} T )( 1 + p^{4} T ) \)
29 \( ( 1 - p^{4} T )( 1 + p^{4} T ) \)
31 \( 1 - 583439 T + p^{8} T^{2} \)
37 \( 1 - 503522 T + p^{8} T^{2} \)
41 \( ( 1 - p^{4} T )( 1 + p^{4} T ) \)
43 \( 1 - 3344879 T + p^{8} T^{2} \)
47 \( ( 1 - p^{4} T )( 1 + p^{4} T ) \)
53 \( ( 1 - p^{4} T )( 1 + p^{4} T ) \)
59 \( ( 1 - p^{4} T )( 1 + p^{4} T ) \)
61 \( 1 - 24133919 T + p^{8} T^{2} \)
67 \( 1 - 37296239 T + p^{8} T^{2} \)
71 \( ( 1 - p^{4} T )( 1 + p^{4} T ) \)
73 \( 1 - 16169282 T + p^{8} T^{2} \)
79 \( 1 + 18887038 T + p^{8} T^{2} \)
83 \( ( 1 - p^{4} T )( 1 + p^{4} T ) \)
89 \( ( 1 - p^{4} T )( 1 + p^{4} T ) \)
97 \( 1 + 94775521 T + p^{8} T^{2} \)
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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

−10.04505463951778579049226672843, −9.457812558833705499932708867167, −8.401761480301838353399774150909, −7.60860803114809691525833758219, −6.70124771773656057777228184492, −5.25308503247455011493072546598, −4.17684434029253983358912484936, −3.06088957976751868315680744958, −2.10519632326143796356731070066, −0.845530809961226658113449772068, 0.845530809961226658113449772068, 2.10519632326143796356731070066, 3.06088957976751868315680744958, 4.17684434029253983358912484936, 5.25308503247455011493072546598, 6.70124771773656057777228184492, 7.60860803114809691525833758219, 8.401761480301838353399774150909, 9.457812558833705499932708867167, 10.04505463951778579049226672843

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