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Properties

Label 2-150-3.2-c2-0-2
Degree $2$
Conductor $150$
Sign $-0.981 + 0.193i$
Analytic cond. $4.08720$
Root an. cond. $2.02168$
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  + 1.41i·2-s + (0.581 + 2.94i)3-s − 2.00·4-s + (−4.16 + 0.821i)6-s − 11.4·7-s − 2.82i·8-s + (−8.32 + 3.42i)9-s + 8.48i·11-s + (−1.16 − 5.88i)12-s + 10·13-s − 16.2i·14-s + 4.00·16-s + 3.55i·17-s + (−4.83 − 11.7i)18-s − 10.9·19-s + ⋯
L(s)  = 1  + 0.707i·2-s + (0.193 + 0.981i)3-s − 0.500·4-s + (−0.693 + 0.136i)6-s − 1.64·7-s − 0.353i·8-s + (−0.924 + 0.380i)9-s + 0.771i·11-s + (−0.0968 − 0.490i)12-s + 0.769·13-s − 1.16i·14-s + 0.250·16-s + 0.209i·17-s + (−0.268 − 0.654i)18-s − 0.577·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.981 + 0.193i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.981 + 0.193i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(150\)    =    \(2 \cdot 3 \cdot 5^{2}\)
Sign: $-0.981 + 0.193i$
Analytic conductor: \(4.08720\)
Root analytic conductor: \(2.02168\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{150} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 150,\ (\ :1),\ -0.981 + 0.193i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0837619 - 0.856614i\)
\(L(\frac12)\) \(\approx\) \(0.0837619 - 0.856614i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 1.41iT \)
3 \( 1 + (-0.581 - 2.94i)T \)
5 \( 1 \)
good7 \( 1 + 11.4T + 49T^{2} \)
11 \( 1 - 8.48iT - 121T^{2} \)
13 \( 1 - 10T + 169T^{2} \)
17 \( 1 - 3.55iT - 289T^{2} \)
19 \( 1 + 10.9T + 361T^{2} \)
23 \( 1 - 17.6iT - 529T^{2} \)
29 \( 1 - 26.8iT - 841T^{2} \)
31 \( 1 - 8T + 961T^{2} \)
37 \( 1 - 59.9T + 1.36e3T^{2} \)
41 \( 1 - 20.5iT - 1.68e3T^{2} \)
43 \( 1 + 42.4T + 1.84e3T^{2} \)
47 \( 1 - 88.2iT - 2.20e3T^{2} \)
53 \( 1 - 3.55iT - 2.80e3T^{2} \)
59 \( 1 + 77.7iT - 3.48e3T^{2} \)
61 \( 1 - 21.9T + 3.72e3T^{2} \)
67 \( 1 - 53.5T + 4.48e3T^{2} \)
71 \( 1 - 69.2iT - 5.04e3T^{2} \)
73 \( 1 + 12.0T + 5.32e3T^{2} \)
79 \( 1 + 9.02T + 6.24e3T^{2} \)
83 \( 1 - 0.688iT - 6.88e3T^{2} \)
89 \( 1 - 7.10iT - 7.92e3T^{2} \)
97 \( 1 + 111.T + 9.40e3T^{2} \)
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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

−13.36697219077053081871227101038, −12.61863275210969286798698582389, −11.06503354128551962418472812375, −9.873455809937881274816424548907, −9.389355203860559621571802809411, −8.217702380139115850003071541153, −6.75186185952701245927610071417, −5.79527391303426034787284621494, −4.33046238448489360711156976066, −3.18293536653060730964876393940, 0.53088999045856781984652839615, 2.58777466674287132775749809878, 3.70279191790099180119027234991, 5.91482524787002236173861039855, 6.69961104362695071588036372569, 8.238570670487706739400364426427, 9.127663375008786060121927296887, 10.25813089940548252555057995317, 11.42775555828861032974478512776, 12.36824995436502367760488364641

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