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Properties

Label 2-30e2-45.29-c2-0-14
Degree $2$
Conductor $900$
Sign $0.595 - 0.803i$
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  + (2.59 + 1.5i)3-s + (−3.24 − 1.87i)7-s + (4.5 + 7.79i)9-s + (−10.1 − 5.84i)11-s + (1.95 − 1.12i)13-s + 11.6·17-s + 26.7·19-s + (−5.61 − 9.73i)21-s + (9.95 + 17.2i)23-s + 27i·27-s + (38.2 + 22.0i)29-s + (26.1 + 45.2i)31-s + (−17.5 − 30.3i)33-s + 14i·37-s + 6.76·39-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)3-s + (−0.463 − 0.267i)7-s + (0.5 + 0.866i)9-s + (−0.919 − 0.531i)11-s + (0.150 − 0.0866i)13-s + 0.687·17-s + 1.40·19-s + (−0.267 − 0.463i)21-s + (0.432 + 0.749i)23-s + i·27-s + (1.31 + 0.761i)29-s + (0.842 + 1.45i)31-s + (−0.531 − 0.919i)33-s + 0.378i·37-s + 0.173·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.595 - 0.803i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.595 - 0.803i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.595 - 0.803i$
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: \((2,\ 900,\ (\ :1),\ 0.595 - 0.803i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.455548996\)
\(L(\frac12)\) \(\approx\) \(2.455548996\)
\(L(2)\) 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 \( 1 + (-2.59 - 1.5i)T \)
5 \( 1 \)
good7 \( 1 + (3.24 + 1.87i)T + (24.5 + 42.4i)T^{2} \)
11 \( 1 + (10.1 + 5.84i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (-1.95 + 1.12i)T + (84.5 - 146. i)T^{2} \)
17 \( 1 - 11.6T + 289T^{2} \)
19 \( 1 - 26.7T + 361T^{2} \)
23 \( 1 + (-9.95 - 17.2i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + (-38.2 - 22.0i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (-26.1 - 45.2i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 - 14iT - 1.36e3T^{2} \)
41 \( 1 + (22.5 - 12.9i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-36.3 - 20.9i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (22.7 - 39.3i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 - 10.8T + 2.80e3T^{2} \)
59 \( 1 + (-31.8 + 18.4i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-22.6 + 39.1i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (86.5 - 49.9i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 102. iT - 5.04e3T^{2} \)
73 \( 1 - 13.7iT - 5.32e3T^{2} \)
79 \( 1 + (-28.3 + 49.1i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (45.0 - 78i)T + (-3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + 95.2iT - 7.92e3T^{2} \)
97 \( 1 + (-88.0 - 50.8i)T + (4.70e3 + 8.14e3i)T^{2} \)
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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

−10.05510310918325393358796202247, −9.250444191639593271193632166354, −8.339989049540933025269793178811, −7.69960948611534930857262923628, −6.77241971967536068701757610547, −5.46136038674514932364285164045, −4.73115110211025264940437532370, −3.24179543834265442627308076118, −3.04260566198059395781726596570, −1.22571277320524442262000080834, 0.830536382739507228732134861132, 2.36958609519045993893076349358, 3.07160776117176220826866322952, 4.27477584657144763758180363563, 5.47774493239306762388083033821, 6.48532158783619797859683768903, 7.41614238258892428147093988734, 8.019826336330265343888302845928, 8.907670102006679077991965865099, 9.805785181308572847162390309916

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