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Properties

Label 2-30e2-36.23-c1-0-103
Degree $2$
Conductor $900$
Sign $-0.341 - 0.940i$
Analytic cond. $7.18653$
Root an. cond. $2.68077$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.287 − 1.38i)2-s + (−0.248 − 1.71i)3-s + (−1.83 + 0.795i)4-s + (−2.30 + 0.836i)6-s + (1.48 − 0.859i)7-s + (1.62 + 2.31i)8-s + (−2.87 + 0.852i)9-s + (−2.82 − 4.90i)11-s + (1.82 + 2.94i)12-s + (0.463 − 0.802i)13-s + (−1.61 − 1.81i)14-s + (2.73 − 2.91i)16-s − 4.20i·17-s + (2.00 + 3.73i)18-s − 0.00833i·19-s + ⋯
L(s)  = 1  + (−0.203 − 0.979i)2-s + (−0.143 − 0.989i)3-s + (−0.917 + 0.397i)4-s + (−0.939 + 0.341i)6-s + (0.562 − 0.324i)7-s + (0.575 + 0.817i)8-s + (−0.958 + 0.284i)9-s + (−0.853 − 1.47i)11-s + (0.525 + 0.850i)12-s + (0.128 − 0.222i)13-s + (−0.432 − 0.484i)14-s + (0.683 − 0.729i)16-s − 1.02i·17-s + (0.472 + 0.881i)18-s − 0.00191i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.341 - 0.940i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.341 - 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.341 - 0.940i$
Analytic conductor: \(7.18653\)
Root analytic conductor: \(2.68077\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{900} (851, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 900,\ (\ :1/2),\ -0.341 - 0.940i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.340512 + 0.485829i\)
\(L(\frac12)\) \(\approx\) \(0.340512 + 0.485829i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.287 + 1.38i)T \)
3 \( 1 + (0.248 + 1.71i)T \)
5 \( 1 \)
good7 \( 1 + (-1.48 + 0.859i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.82 + 4.90i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.463 + 0.802i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 4.20iT - 17T^{2} \)
19 \( 1 + 0.00833iT - 19T^{2} \)
23 \( 1 + (3.90 - 6.76i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (7.85 - 4.53i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (4.83 + 2.79i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 1.71T + 37T^{2} \)
41 \( 1 + (-3.20 - 1.84i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-5.07 + 2.92i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (2.19 + 3.80i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 0.509iT - 53T^{2} \)
59 \( 1 + (-3.39 + 5.88i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.202 - 0.350i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.72 - 2.14i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 7.65T + 71T^{2} \)
73 \( 1 - 8.61T + 73T^{2} \)
79 \( 1 + (13.5 - 7.83i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (4.17 + 7.23i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 12.5iT - 89T^{2} \)
97 \( 1 + (-8.46 - 14.6i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.525529222119412873392226501657, −8.675358278238758688853534157352, −7.83444859350575128283844617579, −7.38851927033737627976740983503, −5.75694454766064543838834954718, −5.25687502867444716752759916990, −3.71598438523406373981975306678, −2.76850968512125065849182767729, −1.57633945841274844020885761848, −0.30779217573926401655228288968, 2.13943562022286113735793594609, 3.97369150402912604375582521234, 4.60464930695545786659815116358, 5.47002196564411847905815603245, 6.24194375185161777466819482068, 7.44176350103130371530463198557, 8.181886065075886394424093705393, 8.968354709444764647267315548529, 9.813723782752176278584988871301, 10.39111351396735012084021897328

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