Properties

Label 2-30e2-4.3-c2-0-9
Degree $2$
Conductor $900$
Sign $-0.5 + 0.866i$
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  + (1 + 1.73i)2-s + (−1.99 + 3.46i)4-s + 10.3i·7-s − 7.99·8-s + 10.3i·11-s − 18·13-s + (−18 + 10.3i)14-s + (−8 − 13.8i)16-s + 10·17-s − 13.8i·19-s + (−18 + 10.3i)22-s − 6.92i·23-s + (−18 − 31.1i)26-s + (−36 − 20.7i)28-s + 36·29-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)2-s + (−0.499 + 0.866i)4-s + 1.48i·7-s − 0.999·8-s + 0.944i·11-s − 1.38·13-s + (−1.28 + 0.742i)14-s + (−0.5 − 0.866i)16-s + 0.588·17-s − 0.729i·19-s + (−0.818 + 0.472i)22-s − 0.301i·23-s + (−0.692 − 1.19i)26-s + (−1.28 − 0.742i)28-s + 1.24·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.5 + 0.866i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.5 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.5 + 0.866i$
Analytic conductor: \(24.5232\)
Root analytic conductor: \(4.95209\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{900} (451, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 900,\ (\ :1),\ -0.5 + 0.866i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.005180289\)
\(L(\frac12)\) \(\approx\) \(1.005180289\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1 - 1.73i)T \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 - 10.3iT - 49T^{2} \)
11 \( 1 - 10.3iT - 121T^{2} \)
13 \( 1 + 18T + 169T^{2} \)
17 \( 1 - 10T + 289T^{2} \)
19 \( 1 + 13.8iT - 361T^{2} \)
23 \( 1 + 6.92iT - 529T^{2} \)
29 \( 1 - 36T + 841T^{2} \)
31 \( 1 - 6.92iT - 961T^{2} \)
37 \( 1 + 54T + 1.36e3T^{2} \)
41 \( 1 + 18T + 1.68e3T^{2} \)
43 \( 1 - 20.7iT - 1.84e3T^{2} \)
47 \( 1 - 2.20e3T^{2} \)
53 \( 1 + 26T + 2.80e3T^{2} \)
59 \( 1 + 31.1iT - 3.48e3T^{2} \)
61 \( 1 + 74T + 3.72e3T^{2} \)
67 \( 1 - 41.5iT - 4.48e3T^{2} \)
71 \( 1 + 103. iT - 5.04e3T^{2} \)
73 \( 1 + 36T + 5.32e3T^{2} \)
79 \( 1 + 90.0iT - 6.24e3T^{2} \)
83 \( 1 - 90.0iT - 6.88e3T^{2} \)
89 \( 1 - 18T + 7.92e3T^{2} \)
97 \( 1 - 72T + 9.40e3T^{2} \)
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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

−10.23089801128313510943441740399, −9.387976789205114897970052659484, −8.740590340222497032252455340999, −7.81409981894127600642898685582, −6.99910128620688543507556032620, −6.16778990892014727661048587078, −5.06793166461503951398854127958, −4.75263145445140604838458051010, −3.14069843552697171708698947382, −2.25712449346531607530462782551, 0.27263540512954913450864968701, 1.44138269675064367895320038094, 2.93957805555843669080057086622, 3.78005164997067597649200975304, 4.69765888267163942743976470368, 5.60758832718049720137127617911, 6.71899796186069682518214152659, 7.63145445143976453830458523809, 8.628667826421034459072840981055, 9.801281212016367616404375153480

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