Properties

Label 2-30e2-3.2-c2-0-9
Degree $2$
Conductor $900$
Sign $-0.577 + 0.816i$
Analytic cond. $24.5232$
Root an. cond. $4.95209$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 7-s + 4.24i·11-s − 7·13-s − 4.24i·17-s − 7·19-s − 29.6i·23-s − 29.6i·29-s + 17·31-s − 16·37-s + 50.9i·41-s − 55·43-s − 46.6i·47-s − 48·49-s − 84.8i·53-s − 55.1i·59-s + ⋯
L(s)  = 1  − 0.142·7-s + 0.385i·11-s − 0.538·13-s − 0.249i·17-s − 0.368·19-s − 1.29i·23-s − 1.02i·29-s + 0.548·31-s − 0.432·37-s + 1.24i·41-s − 1.27·43-s − 0.992i·47-s − 0.979·49-s − 1.60i·53-s − 0.934i·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.577 + 0.816i$
Analytic conductor: \(24.5232\)
Root analytic conductor: \(4.95209\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{900} (701, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 900,\ (\ :1),\ -0.577 + 0.816i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.7960827985\)
\(L(\frac12)\) \(\approx\) \(0.7960827985\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + T + 49T^{2} \)
11 \( 1 - 4.24iT - 121T^{2} \)
13 \( 1 + 7T + 169T^{2} \)
17 \( 1 + 4.24iT - 289T^{2} \)
19 \( 1 + 7T + 361T^{2} \)
23 \( 1 + 29.6iT - 529T^{2} \)
29 \( 1 + 29.6iT - 841T^{2} \)
31 \( 1 - 17T + 961T^{2} \)
37 \( 1 + 16T + 1.36e3T^{2} \)
41 \( 1 - 50.9iT - 1.68e3T^{2} \)
43 \( 1 + 55T + 1.84e3T^{2} \)
47 \( 1 + 46.6iT - 2.20e3T^{2} \)
53 \( 1 + 84.8iT - 2.80e3T^{2} \)
59 \( 1 + 55.1iT - 3.48e3T^{2} \)
61 \( 1 - 65T + 3.72e3T^{2} \)
67 \( 1 + 49T + 4.48e3T^{2} \)
71 \( 1 - 50.9iT - 5.04e3T^{2} \)
73 \( 1 + 88T + 5.32e3T^{2} \)
79 \( 1 + 40T + 6.24e3T^{2} \)
83 \( 1 + 156. iT - 6.88e3T^{2} \)
89 \( 1 + 101. iT - 7.92e3T^{2} \)
97 \( 1 - 41T + 9.40e3T^{2} \)
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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

−9.873532653723532485609668440358, −8.698942946515618136103189578833, −8.026068277993017347704229638670, −6.95618777685981261028276547107, −6.30727035105721454465504149852, −5.08554236692868200805494491873, −4.32692976011623481158418626891, −3.05219076596568358860490604859, −1.95701962426430871495074973759, −0.25413368709989913992283162752, 1.44155157006035149366648738820, 2.82307665051756833932936444782, 3.83394919632874867023228838620, 4.97677967019516916225446648597, 5.84316269493855837198368425176, 6.83270660119798512929282084297, 7.64503409001345623449333398466, 8.583328510590130601161354482856, 9.351530177106777895322213775939, 10.21905658310920424624800611829

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