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Properties

Label 2-30e2-5.4-c3-0-12
Degree $2$
Conductor $900$
Sign $0.894 + 0.447i$
Analytic cond. $53.1017$
Root an. cond. $7.28709$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 7i·7-s + 54·11-s − 55i·13-s + 18i·17-s + 25·19-s + 18i·23-s − 54·29-s − 271·31-s − 314i·37-s + 360·41-s − 163i·43-s + 522i·47-s + 294·49-s + 36i·53-s + 126·59-s + ⋯
L(s)  = 1  + 0.377i·7-s + 1.48·11-s − 1.17i·13-s + 0.256i·17-s + 0.301·19-s + 0.163i·23-s − 0.345·29-s − 1.57·31-s − 1.39i·37-s + 1.37·41-s − 0.578i·43-s + 1.62i·47-s + 0.857·49-s + 0.0933i·53-s + 0.278·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.894 + 0.447i$
Analytic conductor: \(53.1017\)
Root analytic conductor: \(7.28709\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{900} (649, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 900,\ (\ :3/2),\ 0.894 + 0.447i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.196241152\)
\(L(\frac12)\) \(\approx\) \(2.196241152\)
\(L(\frac{5}{2})\) 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 \( 1 - 7iT - 343T^{2} \)
11 \( 1 - 54T + 1.33e3T^{2} \)
13 \( 1 + 55iT - 2.19e3T^{2} \)
17 \( 1 - 18iT - 4.91e3T^{2} \)
19 \( 1 - 25T + 6.85e3T^{2} \)
23 \( 1 - 18iT - 1.21e4T^{2} \)
29 \( 1 + 54T + 2.43e4T^{2} \)
31 \( 1 + 271T + 2.97e4T^{2} \)
37 \( 1 + 314iT - 5.06e4T^{2} \)
41 \( 1 - 360T + 6.89e4T^{2} \)
43 \( 1 + 163iT - 7.95e4T^{2} \)
47 \( 1 - 522iT - 1.03e5T^{2} \)
53 \( 1 - 36iT - 1.48e5T^{2} \)
59 \( 1 - 126T + 2.05e5T^{2} \)
61 \( 1 - 47T + 2.26e5T^{2} \)
67 \( 1 - 343iT - 3.00e5T^{2} \)
71 \( 1 - 1.08e3T + 3.57e5T^{2} \)
73 \( 1 + 1.05e3iT - 3.89e5T^{2} \)
79 \( 1 - 568T + 4.93e5T^{2} \)
83 \( 1 + 1.42e3iT - 5.71e5T^{2} \)
89 \( 1 - 1.44e3T + 7.04e5T^{2} \)
97 \( 1 - 439iT - 9.12e5T^{2} \)
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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

−9.378385635052092194339047562209, −9.082140258860466943296266022024, −7.923206977798691821157488472972, −7.17845278542478162134890350893, −6.06772443958902467981983828011, −5.45849187789021591257034664522, −4.15261123177848659814572961117, −3.31631711913275901123032187752, −1.99150276417406948795029389390, −0.71826212018856001737328380193, 0.978311506715119554673637558918, 2.08055695867881172962423382344, 3.61310313425532991933151187177, 4.25460544231362368151426102862, 5.40316328573450888373305853268, 6.58937134875236234111564721511, 7.01019777164308631751311503509, 8.130800541329968481664752096672, 9.214571536136264920504379289044, 9.492121335054437903208358807526

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