Properties

Label 2-30e2-1.1-c5-0-15
Degree $2$
Conductor $900$
Sign $1$
Analytic cond. $144.345$
Root an. cond. $12.0143$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 133.·7-s + 345.·11-s − 405.·13-s + 2.13e3·17-s − 1.31e3·19-s + 808.·23-s − 984.·29-s + 8.01e3·31-s + 9.34e3·37-s − 1.91e4·41-s + 5.83e3·43-s + 8.89e3·47-s + 920.·49-s + 3.05e4·53-s − 4.90e4·59-s − 423.·61-s + 1.04e4·67-s + 7.81e3·71-s − 2.71e4·73-s + 4.60e4·77-s + 3.15e4·79-s + 8.75e3·83-s + 2.15e4·89-s − 5.39e4·91-s − 5.26e4·97-s − 7.28e4·101-s + 2.86e3·103-s + ⋯
L(s)  = 1  + 1.02·7-s + 0.861·11-s − 0.664·13-s + 1.78·17-s − 0.835·19-s + 0.318·23-s − 0.217·29-s + 1.49·31-s + 1.12·37-s − 1.77·41-s + 0.481·43-s + 0.587·47-s + 0.0547·49-s + 1.49·53-s − 1.83·59-s − 0.0145·61-s + 0.284·67-s + 0.184·71-s − 0.596·73-s + 0.884·77-s + 0.569·79-s + 0.139·83-s + 0.288·89-s − 0.682·91-s − 0.568·97-s − 0.711·101-s + 0.0265·103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(144.345\)
Root analytic conductor: \(12.0143\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 900,\ (\ :5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(2.977933841\)
\(L(\frac12)\) \(\approx\) \(2.977933841\)
\(L(\frac{7}{2})\) 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 \( 1 \)
5 \( 1 \)
good7 \( 1 - 133.T + 1.68e4T^{2} \)
11 \( 1 - 345.T + 1.61e5T^{2} \)
13 \( 1 + 405.T + 3.71e5T^{2} \)
17 \( 1 - 2.13e3T + 1.41e6T^{2} \)
19 \( 1 + 1.31e3T + 2.47e6T^{2} \)
23 \( 1 - 808.T + 6.43e6T^{2} \)
29 \( 1 + 984.T + 2.05e7T^{2} \)
31 \( 1 - 8.01e3T + 2.86e7T^{2} \)
37 \( 1 - 9.34e3T + 6.93e7T^{2} \)
41 \( 1 + 1.91e4T + 1.15e8T^{2} \)
43 \( 1 - 5.83e3T + 1.47e8T^{2} \)
47 \( 1 - 8.89e3T + 2.29e8T^{2} \)
53 \( 1 - 3.05e4T + 4.18e8T^{2} \)
59 \( 1 + 4.90e4T + 7.14e8T^{2} \)
61 \( 1 + 423.T + 8.44e8T^{2} \)
67 \( 1 - 1.04e4T + 1.35e9T^{2} \)
71 \( 1 - 7.81e3T + 1.80e9T^{2} \)
73 \( 1 + 2.71e4T + 2.07e9T^{2} \)
79 \( 1 - 3.15e4T + 3.07e9T^{2} \)
83 \( 1 - 8.75e3T + 3.93e9T^{2} \)
89 \( 1 - 2.15e4T + 5.58e9T^{2} \)
97 \( 1 + 5.26e4T + 8.58e9T^{2} \)
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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

−9.412306862755829076504562939737, −8.396035552642305373496387610299, −7.79953548364807440087310088854, −6.86407820183503947915249388670, −5.85540346152618810663474225536, −4.92817251129582366108317100660, −4.11325681819891330154035832542, −2.93187651098708002583083257459, −1.72108598203957642866515154245, −0.816249180844321590697400541328, 0.816249180844321590697400541328, 1.72108598203957642866515154245, 2.93187651098708002583083257459, 4.11325681819891330154035832542, 4.92817251129582366108317100660, 5.85540346152618810663474225536, 6.86407820183503947915249388670, 7.79953548364807440087310088854, 8.396035552642305373496387610299, 9.412306862755829076504562939737

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