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Properties

Label 2-30e2-100.19-c0-0-0
Degree $2$
Conductor $900$
Sign $0.968 + 0.248i$
Analytic cond. $0.449158$
Root an. cond. $0.670192$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.951 + 0.309i)2-s + (0.809 − 0.587i)4-s + (0.587 − 0.809i)5-s + (−0.587 + 0.809i)8-s + (−0.309 + 0.951i)10-s + (1.11 + 0.363i)13-s + (0.309 − 0.951i)16-s + (−0.363 + 0.5i)17-s − 0.999i·20-s + (−0.309 − 0.951i)25-s − 1.17·26-s + (1.53 − 1.11i)29-s + i·32-s + (0.190 − 0.587i)34-s + (−1.80 − 0.587i)37-s + ⋯
L(s)  = 1  + (−0.951 + 0.309i)2-s + (0.809 − 0.587i)4-s + (0.587 − 0.809i)5-s + (−0.587 + 0.809i)8-s + (−0.309 + 0.951i)10-s + (1.11 + 0.363i)13-s + (0.309 − 0.951i)16-s + (−0.363 + 0.5i)17-s − 0.999i·20-s + (−0.309 − 0.951i)25-s − 1.17·26-s + (1.53 − 1.11i)29-s + i·32-s + (0.190 − 0.587i)34-s + (−1.80 − 0.587i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 + 0.248i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 + 0.248i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.968 + 0.248i$
Analytic conductor: \(0.449158\)
Root analytic conductor: \(0.670192\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{900} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 900,\ (\ :0),\ 0.968 + 0.248i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7570795364\)
\(L(\frac12)\) \(\approx\) \(0.7570795364\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.951 - 0.309i)T \)
3 \( 1 \)
5 \( 1 + (-0.587 + 0.809i)T \)
good7 \( 1 + T^{2} \)
11 \( 1 + (0.809 - 0.587i)T^{2} \)
13 \( 1 + (-1.11 - 0.363i)T + (0.809 + 0.587i)T^{2} \)
17 \( 1 + (0.363 - 0.5i)T + (-0.309 - 0.951i)T^{2} \)
19 \( 1 + (-0.309 - 0.951i)T^{2} \)
23 \( 1 + (-0.809 + 0.587i)T^{2} \)
29 \( 1 + (-1.53 + 1.11i)T + (0.309 - 0.951i)T^{2} \)
31 \( 1 + (-0.309 - 0.951i)T^{2} \)
37 \( 1 + (1.80 + 0.587i)T + (0.809 + 0.587i)T^{2} \)
41 \( 1 + (-0.363 + 1.11i)T + (-0.809 - 0.587i)T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + (0.309 - 0.951i)T^{2} \)
53 \( 1 + (-0.951 - 1.30i)T + (-0.309 + 0.951i)T^{2} \)
59 \( 1 + (0.809 + 0.587i)T^{2} \)
61 \( 1 + (-0.5 - 1.53i)T + (-0.809 + 0.587i)T^{2} \)
67 \( 1 + (0.309 + 0.951i)T^{2} \)
71 \( 1 + (-0.309 + 0.951i)T^{2} \)
73 \( 1 + (-1.11 + 0.363i)T + (0.809 - 0.587i)T^{2} \)
79 \( 1 + (-0.309 + 0.951i)T^{2} \)
83 \( 1 + (0.309 + 0.951i)T^{2} \)
89 \( 1 + (-0.587 - 1.80i)T + (-0.809 + 0.587i)T^{2} \)
97 \( 1 + (1.11 + 1.53i)T + (-0.309 + 0.951i)T^{2} \)
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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

−10.21826839466972879645532036690, −9.231227054363286902934369693906, −8.685973993212821283784075985978, −8.054390145677978122561668408333, −6.83186969632131026396116981700, −6.10914631940864261500616416009, −5.29725196833055762510460688585, −4.03858562433050211447302537035, −2.36743299275316747840188736773, −1.22645270089649264872582687126, 1.50391139544768891975352831776, 2.78073571167309600716951185102, 3.57903040596588651087760689469, 5.23045606078875294149595260070, 6.50956659816960395212163711268, 6.79794082271213732748881607671, 8.036486771068602534354189322714, 8.706954756493413026002513073915, 9.625478852241029843051040886461, 10.34216056191007270820809325809

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