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Properties

Label 2-30e2-45.29-c2-0-12
Degree $2$
Conductor $900$
Sign $0.0297 - 0.999i$
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  + (1.60 + 2.53i)3-s + (−8.13 − 4.69i)7-s + (−3.82 + 8.14i)9-s + (15.4 + 8.90i)11-s + (11.9 − 6.92i)13-s + 30.7·17-s − 28.5·19-s + (−1.19 − 28.1i)21-s + (6.16 + 10.6i)23-s + (−26.7 + 3.42i)27-s + (5.02 + 2.90i)29-s + (−3.25 − 5.63i)31-s + (2.26 + 53.3i)33-s + 66.5i·37-s + (36.8 + 19.2i)39-s + ⋯
L(s)  = 1  + (0.536 + 0.844i)3-s + (−1.16 − 0.670i)7-s + (−0.424 + 0.905i)9-s + (1.40 + 0.809i)11-s + (0.922 − 0.532i)13-s + 1.80·17-s − 1.50·19-s + (−0.0568 − 1.34i)21-s + (0.267 + 0.464i)23-s + (−0.991 + 0.126i)27-s + (0.173 + 0.100i)29-s + (−0.104 − 0.181i)31-s + (0.0686 + 1.61i)33-s + 1.79i·37-s + (0.943 + 0.492i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0297 - 0.999i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.0297 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.0297 - 0.999i$
Analytic conductor: \(24.5232\)
Root analytic conductor: \(4.95209\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{900} (749, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 900,\ (\ :1),\ 0.0297 - 0.999i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.066954594\)
\(L(\frac12)\) \(\approx\) \(2.066954594\)
\(L(2)\) 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 \( 1 + (-1.60 - 2.53i)T \)
5 \( 1 \)
good7 \( 1 + (8.13 + 4.69i)T + (24.5 + 42.4i)T^{2} \)
11 \( 1 + (-15.4 - 8.90i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (-11.9 + 6.92i)T + (84.5 - 146. i)T^{2} \)
17 \( 1 - 30.7T + 289T^{2} \)
19 \( 1 + 28.5T + 361T^{2} \)
23 \( 1 + (-6.16 - 10.6i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + (-5.02 - 2.90i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (3.25 + 5.63i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 - 66.5iT - 1.36e3T^{2} \)
41 \( 1 + (33.0 - 19.0i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-47.7 - 27.5i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (8.23 - 14.2i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 - 69.8T + 2.80e3T^{2} \)
59 \( 1 + (91.2 - 52.6i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (33.5 - 58.1i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-39.8 + 22.9i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 31.1iT - 5.04e3T^{2} \)
73 \( 1 - 73.5iT - 5.32e3T^{2} \)
79 \( 1 + (-47.3 + 81.9i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (-7.73 + 13.4i)T + (-3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 - 52.7iT - 7.92e3T^{2} \)
97 \( 1 + (66.0 + 38.1i)T + (4.70e3 + 8.14e3i)T^{2} \)
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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

−10.05584067604964565461609601797, −9.435116881173147700194194151191, −8.605457504438855582668479317738, −7.66855720677961566880069609628, −6.64439182316626499635102912592, −5.85043514448313088392917859656, −4.48790540760737836842705596655, −3.73117306814431835278416243702, −3.03777739596203819592947784607, −1.29478598879223699348507647632, 0.70264840254697542424549333643, 2.00429558468746977860635192328, 3.30449250190960204481099769378, 3.84724983260107607104093311883, 5.81240442121319994041027720441, 6.26791313289228749697855470953, 6.98348292008820824431211655777, 8.176408296543460007918625782988, 8.974851436844050157696668028515, 9.278091205694471318652443468043

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