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Properties

Label 2-30e2-4.3-c0-0-0
Degree $2$
Conductor $900$
Sign $1$
Analytic cond. $0.449158$
Root an. cond. $0.670192$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s − 8-s + 16-s + 2·17-s − 32-s − 2·34-s + 49-s + 2·53-s − 2·61-s + 64-s + 2·68-s − 98-s − 2·106-s − 2·109-s − 2·113-s + ⋯
L(s)  = 1  − 2-s + 4-s − 8-s + 16-s + 2·17-s − 32-s − 2·34-s + 49-s + 2·53-s − 2·61-s + 64-s + 2·68-s − 98-s − 2·106-s − 2·109-s − 2·113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(0.449158\)
Root analytic conductor: \(0.670192\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{900} (451, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 900,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6858624574\)
\(L(\frac12)\) \(\approx\) \(0.6858624574\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 \)
5 \( 1 \)
good7 \( ( 1 - T )( 1 + T ) \)
11 \( ( 1 - T )( 1 + T ) \)
13 \( 1 + T^{2} \)
17 \( ( 1 - T )^{2} \)
19 \( ( 1 - T )( 1 + T ) \)
23 \( ( 1 - T )( 1 + T ) \)
29 \( 1 + T^{2} \)
31 \( ( 1 - T )( 1 + T ) \)
37 \( 1 + T^{2} \)
41 \( 1 + T^{2} \)
43 \( ( 1 - T )( 1 + T ) \)
47 \( ( 1 - T )( 1 + T ) \)
53 \( ( 1 - T )^{2} \)
59 \( ( 1 - T )( 1 + T ) \)
61 \( ( 1 + T )^{2} \)
67 \( ( 1 - T )( 1 + T ) \)
71 \( ( 1 - T )( 1 + T ) \)
73 \( 1 + T^{2} \)
79 \( ( 1 - T )( 1 + T ) \)
83 \( ( 1 - T )( 1 + T ) \)
89 \( 1 + T^{2} \)
97 \( 1 + T^{2} \)
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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

−10.21841384456184385415328160901, −9.521515455277577513914396452014, −8.666363099292299297458247504930, −7.81234257808225024594470279604, −7.20355099545403028608573467797, −6.10178498788704571155608264607, −5.32095528141858338131602343817, −3.73968842597621249229728175034, −2.67782062274184195392696417379, −1.24960724593285936378801101482, 1.24960724593285936378801101482, 2.67782062274184195392696417379, 3.73968842597621249229728175034, 5.32095528141858338131602343817, 6.10178498788704571155608264607, 7.20355099545403028608573467797, 7.81234257808225024594470279604, 8.666363099292299297458247504930, 9.521515455277577513914396452014, 10.21841384456184385415328160901

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