Properties

Label 2-30e2-9.2-c2-0-7
Degree $2$
Conductor $900$
Sign $-0.265 - 0.964i$
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  + (−2.65 + 1.39i)3-s + (−1.41 + 2.45i)7-s + (5.10 − 7.40i)9-s + (−0.949 − 0.547i)11-s + (−2.09 − 3.63i)13-s + 7.85i·17-s + 13.3·19-s + (0.340 − 8.48i)21-s + (21.4 − 12.3i)23-s + (−3.23 + 26.8i)27-s + (−2.43 − 1.40i)29-s + (12.0 + 20.8i)31-s + (3.28 + 0.131i)33-s − 49.9·37-s + (10.6 + 6.72i)39-s + ⋯
L(s)  = 1  + (−0.885 + 0.464i)3-s + (−0.202 + 0.350i)7-s + (0.567 − 0.823i)9-s + (−0.0862 − 0.0498i)11-s + (−0.161 − 0.279i)13-s + 0.462i·17-s + 0.703·19-s + (0.0161 − 0.404i)21-s + (0.931 − 0.537i)23-s + (−0.119 + 0.992i)27-s + (−0.0840 − 0.0485i)29-s + (0.389 + 0.673i)31-s + (0.0995 + 0.00398i)33-s − 1.34·37-s + (0.272 + 0.172i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.265 - 0.964i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.265 - 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.265 - 0.964i$
Analytic conductor: \(24.5232\)
Root analytic conductor: \(4.95209\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{900} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 900,\ (\ :1),\ -0.265 - 0.964i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.9657345137\)
\(L(\frac12)\) \(\approx\) \(0.9657345137\)
\(L(2)\) 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 \( 1 + (2.65 - 1.39i)T \)
5 \( 1 \)
good7 \( 1 + (1.41 - 2.45i)T + (-24.5 - 42.4i)T^{2} \)
11 \( 1 + (0.949 + 0.547i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (2.09 + 3.63i)T + (-84.5 + 146. i)T^{2} \)
17 \( 1 - 7.85iT - 289T^{2} \)
19 \( 1 - 13.3T + 361T^{2} \)
23 \( 1 + (-21.4 + 12.3i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (2.43 + 1.40i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (-12.0 - 20.8i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + 49.9T + 1.36e3T^{2} \)
41 \( 1 + (18.9 - 10.9i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-24.5 + 42.4i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (-58.5 - 33.8i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + 49.8iT - 2.80e3T^{2} \)
59 \( 1 + (86.8 - 50.1i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (41.5 - 71.9i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-28.7 - 49.8i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 14.2iT - 5.04e3T^{2} \)
73 \( 1 + 71.5T + 5.32e3T^{2} \)
79 \( 1 + (54.4 - 94.2i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (-69.7 - 40.2i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 - 65.6iT - 7.92e3T^{2} \)
97 \( 1 + (77.4 - 134. i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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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

−10.41491517633445380183464663953, −9.355032965857616293910412646084, −8.705515939253145032180028937776, −7.45288379190552957554208186500, −6.62862335531290853869290531944, −5.69186836478314271645265720815, −5.02562625700416431688255713930, −3.95172701407538804931745314106, −2.82788502984070453312656393736, −1.10255380150099953227340490552, 0.41982329922624372693487516231, 1.70999685515513868228049200119, 3.16248565694616857360694761578, 4.49926821822150701467355347371, 5.31236428971822687990959314504, 6.22777685047178678111614036867, 7.17053303804170974022387333661, 7.61865278856266711182308642462, 8.905442220348203441797077425041, 9.792231124920226882981541924314

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