Properties

Label 2-30e2-1.1-c5-0-17
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  − 53.1·7-s + 585.·11-s + 1.08e3·13-s + 1.59e3·17-s + 2.41e3·19-s − 2.67e3·23-s + 8.43e3·29-s − 5.02e3·31-s − 1.82e3·37-s + 7.97e3·41-s − 1.05e4·43-s − 2.93e4·47-s − 1.39e4·49-s − 2.30e4·53-s − 2.21e3·59-s + 2.19e4·61-s + 410.·67-s + 5.36e4·71-s + 1.75e4·73-s − 3.11e4·77-s + 5.95e4·79-s + 2.29e4·83-s + 5.67e4·89-s − 5.76e4·91-s + 1.51e5·97-s − 1.13e5·101-s − 1.67e4·103-s + ⋯
L(s)  = 1  − 0.409·7-s + 1.45·11-s + 1.78·13-s + 1.33·17-s + 1.53·19-s − 1.05·23-s + 1.86·29-s − 0.938·31-s − 0.219·37-s + 0.740·41-s − 0.870·43-s − 1.94·47-s − 0.831·49-s − 1.12·53-s − 0.0828·59-s + 0.754·61-s + 0.0111·67-s + 1.26·71-s + 0.385·73-s − 0.598·77-s + 1.07·79-s + 0.365·83-s + 0.758·89-s − 0.730·91-s + 1.63·97-s − 1.10·101-s − 0.155·103-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

\(L(3)\) \(\approx\) \(3.081090501\)
\(L(\frac12)\) \(\approx\) \(3.081090501\)
\(L(\frac{7}{2})\) 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 \( 1 + 53.1T + 1.68e4T^{2} \)
11 \( 1 - 585.T + 1.61e5T^{2} \)
13 \( 1 - 1.08e3T + 3.71e5T^{2} \)
17 \( 1 - 1.59e3T + 1.41e6T^{2} \)
19 \( 1 - 2.41e3T + 2.47e6T^{2} \)
23 \( 1 + 2.67e3T + 6.43e6T^{2} \)
29 \( 1 - 8.43e3T + 2.05e7T^{2} \)
31 \( 1 + 5.02e3T + 2.86e7T^{2} \)
37 \( 1 + 1.82e3T + 6.93e7T^{2} \)
41 \( 1 - 7.97e3T + 1.15e8T^{2} \)
43 \( 1 + 1.05e4T + 1.47e8T^{2} \)
47 \( 1 + 2.93e4T + 2.29e8T^{2} \)
53 \( 1 + 2.30e4T + 4.18e8T^{2} \)
59 \( 1 + 2.21e3T + 7.14e8T^{2} \)
61 \( 1 - 2.19e4T + 8.44e8T^{2} \)
67 \( 1 - 410.T + 1.35e9T^{2} \)
71 \( 1 - 5.36e4T + 1.80e9T^{2} \)
73 \( 1 - 1.75e4T + 2.07e9T^{2} \)
79 \( 1 - 5.95e4T + 3.07e9T^{2} \)
83 \( 1 - 2.29e4T + 3.93e9T^{2} \)
89 \( 1 - 5.67e4T + 5.58e9T^{2} \)
97 \( 1 - 1.51e5T + 8.58e9T^{2} \)
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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

−9.482904201195488923640918239792, −8.516538678072063942166773666251, −7.78969783592120984964952814152, −6.55987089487583769620096670384, −6.13652663577266980404974651101, −5.02248036901007447383153584023, −3.65825037686720027475777969154, −3.34484105171854260646095002419, −1.55681833784826077205045242180, −0.878116986483891444923988509612, 0.878116986483891444923988509612, 1.55681833784826077205045242180, 3.34484105171854260646095002419, 3.65825037686720027475777969154, 5.02248036901007447383153584023, 6.13652663577266980404974651101, 6.55987089487583769620096670384, 7.78969783592120984964952814152, 8.516538678072063942166773666251, 9.482904201195488923640918239792

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