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Properties

Label 2-30e2-300.23-c0-0-0
Degree $2$
Conductor $900$
Sign $0.720 - 0.693i$
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  + (0.987 + 0.156i)2-s + (0.951 + 0.309i)4-s + (−0.453 + 0.891i)5-s + (0.891 + 0.453i)8-s + (−0.587 + 0.809i)10-s + (−0.142 − 0.896i)13-s + (0.809 + 0.587i)16-s + (−0.533 + 1.04i)17-s + (−0.707 + 0.707i)20-s + (−0.587 − 0.809i)25-s − 0.907i·26-s + (0.610 − 1.87i)29-s + (0.707 + 0.707i)32-s + (−0.690 + 0.951i)34-s + (−0.309 + 0.0489i)37-s + ⋯
L(s)  = 1  + (0.987 + 0.156i)2-s + (0.951 + 0.309i)4-s + (−0.453 + 0.891i)5-s + (0.891 + 0.453i)8-s + (−0.587 + 0.809i)10-s + (−0.142 − 0.896i)13-s + (0.809 + 0.587i)16-s + (−0.533 + 1.04i)17-s + (−0.707 + 0.707i)20-s + (−0.587 − 0.809i)25-s − 0.907i·26-s + (0.610 − 1.87i)29-s + (0.707 + 0.707i)32-s + (−0.690 + 0.951i)34-s + (−0.309 + 0.0489i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.720 - 0.693i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.720 - 0.693i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.720 - 0.693i$
Analytic conductor: \(0.449158\)
Root analytic conductor: \(0.670192\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{900} (323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 900,\ (\ :0),\ 0.720 - 0.693i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.678575946\)
\(L(\frac12)\) \(\approx\) \(1.678575946\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.987 - 0.156i)T \)
3 \( 1 \)
5 \( 1 + (0.453 - 0.891i)T \)
good7 \( 1 + iT^{2} \)
11 \( 1 + (0.309 - 0.951i)T^{2} \)
13 \( 1 + (0.142 + 0.896i)T + (-0.951 + 0.309i)T^{2} \)
17 \( 1 + (0.533 - 1.04i)T + (-0.587 - 0.809i)T^{2} \)
19 \( 1 + (-0.809 + 0.587i)T^{2} \)
23 \( 1 + (0.951 + 0.309i)T^{2} \)
29 \( 1 + (-0.610 + 1.87i)T + (-0.809 - 0.587i)T^{2} \)
31 \( 1 + (0.809 - 0.587i)T^{2} \)
37 \( 1 + (0.309 - 0.0489i)T + (0.951 - 0.309i)T^{2} \)
41 \( 1 + (1.04 - 1.44i)T + (-0.309 - 0.951i)T^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 + (-0.587 + 0.809i)T^{2} \)
53 \( 1 + (0.863 + 1.69i)T + (-0.587 + 0.809i)T^{2} \)
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 \( 1 + (1.76 + 0.278i)T + (0.951 + 0.309i)T^{2} \)
79 \( 1 + (-0.809 - 0.587i)T^{2} \)
83 \( 1 + (-0.587 - 0.809i)T^{2} \)
89 \( 1 + (-0.253 + 0.183i)T + (0.309 - 0.951i)T^{2} \)
97 \( 1 + (-0.896 - 1.76i)T + (-0.587 + 0.809i)T^{2} \)
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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

−10.50881760726300515051965708608, −9.977947496226990216364125933939, −8.268584491802090880412261008462, −7.87164508951141728687826455595, −6.71119540166231237884624369981, −6.23184802219006896851038236338, −5.10635235600885901129405361651, −4.04742927709698840884975793573, −3.22932875204371312786881908952, −2.17929036591852459450067450580, 1.52614360141319530145078660978, 2.91594584388946809654208969471, 4.09832610009877531153763933818, 4.80071572186294648993526789698, 5.57078962052024463958003650785, 6.82342914285369257813225379827, 7.37004033163099433186580630751, 8.629202309632283597076948980986, 9.305020314635783967264084711832, 10.43217491404989629776185589054

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