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Properties

Label 2-30e2-9.5-c2-0-35
Degree $2$
Conductor $900$
Sign $-0.640 + 0.768i$
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  + (1.92 − 2.30i)3-s + (−0.382 − 0.662i)7-s + (−1.59 − 8.85i)9-s + (6.98 − 4.03i)11-s + (−1.08 + 1.87i)13-s − 24.9i·17-s − 20.7·19-s + (−2.25 − 0.394i)21-s + (14.5 + 8.42i)23-s + (−23.4 − 13.3i)27-s + (−10.1 + 5.86i)29-s + (−0.334 + 0.579i)31-s + (4.16 − 23.8i)33-s + 48.2·37-s + (2.23 + 6.10i)39-s + ⋯
L(s)  = 1  + (0.641 − 0.767i)3-s + (−0.0546 − 0.0946i)7-s + (−0.176 − 0.984i)9-s + (0.635 − 0.366i)11-s + (−0.0833 + 0.144i)13-s − 1.46i·17-s − 1.09·19-s + (−0.107 − 0.0187i)21-s + (0.634 + 0.366i)23-s + (−0.868 − 0.495i)27-s + (−0.350 + 0.202i)29-s + (−0.0107 + 0.0187i)31-s + (0.126 − 0.722i)33-s + 1.30·37-s + (0.0572 + 0.156i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.640 + 0.768i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.640 + 0.768i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.640 + 0.768i$
Analytic conductor: \(24.5232\)
Root analytic conductor: \(4.95209\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{900} (401, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 900,\ (\ :1),\ -0.640 + 0.768i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.889065224\)
\(L(\frac12)\) \(\approx\) \(1.889065224\)
\(L(2)\) 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 \( 1 + (-1.92 + 2.30i)T \)
5 \( 1 \)
good7 \( 1 + (0.382 + 0.662i)T + (-24.5 + 42.4i)T^{2} \)
11 \( 1 + (-6.98 + 4.03i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (1.08 - 1.87i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 + 24.9iT - 289T^{2} \)
19 \( 1 + 20.7T + 361T^{2} \)
23 \( 1 + (-14.5 - 8.42i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (10.1 - 5.86i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (0.334 - 0.579i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 - 48.2T + 1.36e3T^{2} \)
41 \( 1 + (54.1 + 31.2i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (35.6 + 61.7i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-17.5 + 10.1i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 - 82.8iT - 2.80e3T^{2} \)
59 \( 1 + (4.66 + 2.69i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (41.6 + 72.0i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-5.31 + 9.20i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 87.9iT - 5.04e3T^{2} \)
73 \( 1 - 15.5T + 5.32e3T^{2} \)
79 \( 1 + (-16.8 - 29.2i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (48.2 - 27.8i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + 22.3iT - 7.92e3T^{2} \)
97 \( 1 + (-34.4 - 59.7i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−9.260046129475345492005338765801, −8.905305987503869012896337988052, −7.87577618173864157941313152200, −7.04132946243086258209866521947, −6.43759965928015179947581373276, −5.26032915491657745096149092336, −3.99250781146566395608292054135, −3.00839984417725703221079015208, −1.90038842625763418574420892475, −0.55027325055789993838347737892, 1.69054379272734989526313955208, 2.88209098031883429224014125754, 3.99433360342616495392479286939, 4.63145555376480416235916219302, 5.87074717607614262581943722310, 6.76769359257131290252791846088, 7.993387009077608434154875262892, 8.546501146004639265925344146761, 9.404819517245450181556090011805, 10.13571836400658253606361082724

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