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Properties

Label 2-30e2-300.263-c0-0-1
Degree $2$
Conductor $900$
Sign $0.414 - 0.910i$
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  + (0.156 + 0.987i)2-s + (−0.951 + 0.309i)4-s + (0.891 − 0.453i)5-s + (−0.453 − 0.891i)8-s + (0.587 + 0.809i)10-s + (1.76 + 0.278i)13-s + (0.809 − 0.587i)16-s + (−1.04 + 0.533i)17-s + (−0.707 + 0.707i)20-s + (0.587 − 0.809i)25-s + 1.78i·26-s + (0.0966 + 0.297i)29-s + (0.707 + 0.707i)32-s + (−0.690 − 0.951i)34-s + (−0.309 + 1.95i)37-s + ⋯
L(s)  = 1  + (0.156 + 0.987i)2-s + (−0.951 + 0.309i)4-s + (0.891 − 0.453i)5-s + (−0.453 − 0.891i)8-s + (0.587 + 0.809i)10-s + (1.76 + 0.278i)13-s + (0.809 − 0.587i)16-s + (−1.04 + 0.533i)17-s + (−0.707 + 0.707i)20-s + (0.587 − 0.809i)25-s + 1.78i·26-s + (0.0966 + 0.297i)29-s + (0.707 + 0.707i)32-s + (−0.690 − 0.951i)34-s + (−0.309 + 1.95i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.414 - 0.910i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.414 - 0.910i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.414 - 0.910i$
Analytic conductor: \(0.449158\)
Root analytic conductor: \(0.670192\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{900} (863, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 900,\ (\ :0),\ 0.414 - 0.910i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.152744168\)
\(L(\frac12)\) \(\approx\) \(1.152744168\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.156 - 0.987i)T \)
3 \( 1 \)
5 \( 1 + (-0.891 + 0.453i)T \)
good7 \( 1 + iT^{2} \)
11 \( 1 + (0.309 + 0.951i)T^{2} \)
13 \( 1 + (-1.76 - 0.278i)T + (0.951 + 0.309i)T^{2} \)
17 \( 1 + (1.04 - 0.533i)T + (0.587 - 0.809i)T^{2} \)
19 \( 1 + (-0.809 - 0.587i)T^{2} \)
23 \( 1 + (-0.951 + 0.309i)T^{2} \)
29 \( 1 + (-0.0966 - 0.297i)T + (-0.809 + 0.587i)T^{2} \)
31 \( 1 + (0.809 + 0.587i)T^{2} \)
37 \( 1 + (0.309 - 1.95i)T + (-0.951 - 0.309i)T^{2} \)
41 \( 1 + (0.533 + 0.734i)T + (-0.309 + 0.951i)T^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 + (0.587 + 0.809i)T^{2} \)
53 \( 1 + (1.69 + 0.863i)T + (0.587 + 0.809i)T^{2} \)
59 \( 1 + (-0.309 + 0.951i)T^{2} \)
61 \( 1 + (1.53 + 1.11i)T + (0.309 + 0.951i)T^{2} \)
67 \( 1 + (0.587 - 0.809i)T^{2} \)
71 \( 1 + (-0.809 + 0.587i)T^{2} \)
73 \( 1 + (-0.142 - 0.896i)T + (-0.951 + 0.309i)T^{2} \)
79 \( 1 + (-0.809 + 0.587i)T^{2} \)
83 \( 1 + (0.587 - 0.809i)T^{2} \)
89 \( 1 + (-1.59 - 1.16i)T + (0.309 + 0.951i)T^{2} \)
97 \( 1 + (0.278 + 0.142i)T + (0.587 + 0.809i)T^{2} \)
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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

−10.27806617809088493124075167601, −9.353378054353480921367976532972, −8.639364442852992271471443491798, −8.162640236360913201672480780507, −6.63612003352339336720646976819, −6.36383063511847993771795300422, −5.34819804157165677354586926108, −4.47051309866663367470360684068, −3.38825623036177170445340059149, −1.57849827473773916319013860730, 1.46904284801375981271767337577, 2.61660621865662822810301276553, 3.59926867647159677820974377101, 4.69790736560695027299841178247, 5.82195635514294145329020053793, 6.40561592206042947213489246935, 7.78619071324657404583788211471, 8.991983916961511232150284117240, 9.248312181297600262050032034655, 10.48991026934516877344822898162

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