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Properties

Label 2-30e2-300.203-c0-0-0
Degree $2$
Conductor $900$
Sign $0.356 - 0.934i$
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  + (−0.891 + 0.453i)2-s + (0.587 − 0.809i)4-s + (−0.987 − 0.156i)5-s + (−0.156 + 0.987i)8-s + (0.951 − 0.309i)10-s + (−0.896 + 1.76i)13-s + (−0.309 − 0.951i)16-s + (1.87 + 0.297i)17-s + (−0.707 + 0.707i)20-s + (0.951 + 0.309i)25-s − 1.97i·26-s + (1.44 + 1.04i)29-s + (0.707 + 0.707i)32-s + (−1.80 + 0.587i)34-s + (0.809 + 0.412i)37-s + ⋯
L(s)  = 1  + (−0.891 + 0.453i)2-s + (0.587 − 0.809i)4-s + (−0.987 − 0.156i)5-s + (−0.156 + 0.987i)8-s + (0.951 − 0.309i)10-s + (−0.896 + 1.76i)13-s + (−0.309 − 0.951i)16-s + (1.87 + 0.297i)17-s + (−0.707 + 0.707i)20-s + (0.951 + 0.309i)25-s − 1.97i·26-s + (1.44 + 1.04i)29-s + (0.707 + 0.707i)32-s + (−1.80 + 0.587i)34-s + (0.809 + 0.412i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.356 - 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.356 - 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.356 - 0.934i$
Analytic conductor: \(0.449158\)
Root analytic conductor: \(0.670192\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{900} (503, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 900,\ (\ :0),\ 0.356 - 0.934i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5370348359\)
\(L(\frac12)\) \(\approx\) \(0.5370348359\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.891 - 0.453i)T \)
3 \( 1 \)
5 \( 1 + (0.987 + 0.156i)T \)
good7 \( 1 + iT^{2} \)
11 \( 1 + (-0.809 - 0.587i)T^{2} \)
13 \( 1 + (0.896 - 1.76i)T + (-0.587 - 0.809i)T^{2} \)
17 \( 1 + (-1.87 - 0.297i)T + (0.951 + 0.309i)T^{2} \)
19 \( 1 + (0.309 - 0.951i)T^{2} \)
23 \( 1 + (0.587 - 0.809i)T^{2} \)
29 \( 1 + (-1.44 - 1.04i)T + (0.309 + 0.951i)T^{2} \)
31 \( 1 + (-0.309 + 0.951i)T^{2} \)
37 \( 1 + (-0.809 - 0.412i)T + (0.587 + 0.809i)T^{2} \)
41 \( 1 + (0.297 - 0.0966i)T + (0.809 - 0.587i)T^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 + (0.951 - 0.309i)T^{2} \)
53 \( 1 + (1.16 - 0.183i)T + (0.951 - 0.309i)T^{2} \)
59 \( 1 + (0.809 - 0.587i)T^{2} \)
61 \( 1 + (0.363 - 1.11i)T + (-0.809 - 0.587i)T^{2} \)
67 \( 1 + (0.951 + 0.309i)T^{2} \)
71 \( 1 + (0.309 + 0.951i)T^{2} \)
73 \( 1 + (0.278 - 0.142i)T + (0.587 - 0.809i)T^{2} \)
79 \( 1 + (0.309 + 0.951i)T^{2} \)
83 \( 1 + (0.951 + 0.309i)T^{2} \)
89 \( 1 + (0.280 - 0.863i)T + (-0.809 - 0.587i)T^{2} \)
97 \( 1 + (1.76 - 0.278i)T + (0.951 - 0.309i)T^{2} \)
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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

−10.25533520458262273922590206509, −9.577687336697017679990293943432, −8.703030138584828463305306620173, −7.967875545285242817702062677209, −7.21039953847556823222320527991, −6.53103093691612159409917816161, −5.27320367621786531408599916330, −4.36930043618597602290040361502, −2.97849505773504656027527218061, −1.39622071123164531093632587568, 0.800737410943779763670657227828, 2.77181975301344430612239502277, 3.37061683995461990215091699819, 4.68538096177814200061650787726, 5.93040802236101095765148118835, 7.18340668232805453593223807273, 7.943759655829877788392517519113, 8.137920939482265566060771261649, 9.530535003300153030407508728559, 10.14187416871903901768840820021

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