Properties

Label 2-30e2-100.59-c0-0-0
Degree $2$
Conductor $900$
Sign $0.187 + 0.982i$
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  + (−0.587 + 0.809i)2-s + (−0.309 − 0.951i)4-s + (−0.951 − 0.309i)5-s + (0.951 + 0.309i)8-s + (0.809 − 0.587i)10-s + (−1.11 − 1.53i)13-s + (−0.809 + 0.587i)16-s + (−1.53 − 0.5i)17-s + i·20-s + (0.809 + 0.587i)25-s + 1.90·26-s + (−0.363 − 1.11i)29-s i·32-s + (1.30 − 0.951i)34-s + (−0.690 − 0.951i)37-s + ⋯
L(s)  = 1  + (−0.587 + 0.809i)2-s + (−0.309 − 0.951i)4-s + (−0.951 − 0.309i)5-s + (0.951 + 0.309i)8-s + (0.809 − 0.587i)10-s + (−1.11 − 1.53i)13-s + (−0.809 + 0.587i)16-s + (−1.53 − 0.5i)17-s + i·20-s + (0.809 + 0.587i)25-s + 1.90·26-s + (−0.363 − 1.11i)29-s i·32-s + (1.30 − 0.951i)34-s + (−0.690 − 0.951i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.187 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.187 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.187 + 0.982i$
Analytic conductor: \(0.449158\)
Root analytic conductor: \(0.670192\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{900} (559, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 900,\ (\ :0),\ 0.187 + 0.982i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3496794591\)
\(L(\frac12)\) \(\approx\) \(0.3496794591\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.587 - 0.809i)T \)
3 \( 1 \)
5 \( 1 + (0.951 + 0.309i)T \)
good7 \( 1 + T^{2} \)
11 \( 1 + (-0.309 - 0.951i)T^{2} \)
13 \( 1 + (1.11 + 1.53i)T + (-0.309 + 0.951i)T^{2} \)
17 \( 1 + (1.53 + 0.5i)T + (0.809 + 0.587i)T^{2} \)
19 \( 1 + (0.809 + 0.587i)T^{2} \)
23 \( 1 + (0.309 + 0.951i)T^{2} \)
29 \( 1 + (0.363 + 1.11i)T + (-0.809 + 0.587i)T^{2} \)
31 \( 1 + (0.809 + 0.587i)T^{2} \)
37 \( 1 + (0.690 + 0.951i)T + (-0.309 + 0.951i)T^{2} \)
41 \( 1 + (-1.53 + 1.11i)T + (0.309 - 0.951i)T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + (-0.809 + 0.587i)T^{2} \)
53 \( 1 + (-0.587 + 0.190i)T + (0.809 - 0.587i)T^{2} \)
59 \( 1 + (-0.309 + 0.951i)T^{2} \)
61 \( 1 + (-0.5 - 0.363i)T + (0.309 + 0.951i)T^{2} \)
67 \( 1 + (-0.809 - 0.587i)T^{2} \)
71 \( 1 + (0.809 - 0.587i)T^{2} \)
73 \( 1 + (1.11 - 1.53i)T + (-0.309 - 0.951i)T^{2} \)
79 \( 1 + (0.809 - 0.587i)T^{2} \)
83 \( 1 + (-0.809 - 0.587i)T^{2} \)
89 \( 1 + (0.951 + 0.690i)T + (0.309 + 0.951i)T^{2} \)
97 \( 1 + (-1.11 + 0.363i)T + (0.809 - 0.587i)T^{2} \)
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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

−9.988896070691137936149698403746, −9.097343174323470850668379884550, −8.371811039133017427213916724657, −7.54151136700843013506515958967, −7.07582401980829654856950406652, −5.80204430403534948473102132095, −4.96364811490501502751895917208, −4.07732650360711890512433733196, −2.46673674445849908816382040122, −0.40453412435611022830319396204, 1.87223927915976771603717573295, 2.99084047803906362127728988439, 4.18278494522697231599061994211, 4.70481709433412669536379883402, 6.64814945259792471324627068540, 7.17771897198835749228988234300, 8.149813133917595772836141641070, 8.943568426682628284237918771642, 9.609898418471823887152033670553, 10.67539156364736167482402092324

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