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Properties

Label 2-150-5.4-c3-0-2
Degree $2$
Conductor $150$
Sign $0.447 - 0.894i$
Analytic cond. $8.85028$
Root an. cond. $2.97494$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2i·2-s − 3i·3-s − 4·4-s − 6·6-s + 32i·7-s + 8i·8-s − 9·9-s − 60·11-s + 12i·12-s + 34i·13-s + 64·14-s + 16·16-s + 42i·17-s + 18i·18-s + 76·19-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.577i·3-s − 0.5·4-s − 0.408·6-s + 1.72i·7-s + 0.353i·8-s − 0.333·9-s − 1.64·11-s + 0.288i·12-s + 0.725i·13-s + 1.22·14-s + 0.250·16-s + 0.599i·17-s + 0.235i·18-s + 0.917·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(150\)    =    \(2 \cdot 3 \cdot 5^{2}\)
Sign: $0.447 - 0.894i$
Analytic conductor: \(8.85028\)
Root analytic conductor: \(2.97494\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{150} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 150,\ (\ :3/2),\ 0.447 - 0.894i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.700717 + 0.433067i\)
\(L(\frac12)\) \(\approx\) \(0.700717 + 0.433067i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 2iT \)
3 \( 1 + 3iT \)
5 \( 1 \)
good7 \( 1 - 32iT - 343T^{2} \)
11 \( 1 + 60T + 1.33e3T^{2} \)
13 \( 1 - 34iT - 2.19e3T^{2} \)
17 \( 1 - 42iT - 4.91e3T^{2} \)
19 \( 1 - 76T + 6.85e3T^{2} \)
23 \( 1 - 1.21e4T^{2} \)
29 \( 1 + 6T + 2.43e4T^{2} \)
31 \( 1 + 232T + 2.97e4T^{2} \)
37 \( 1 - 134iT - 5.06e4T^{2} \)
41 \( 1 - 234T + 6.89e4T^{2} \)
43 \( 1 - 412iT - 7.95e4T^{2} \)
47 \( 1 + 360iT - 1.03e5T^{2} \)
53 \( 1 + 222iT - 1.48e5T^{2} \)
59 \( 1 + 660T + 2.05e5T^{2} \)
61 \( 1 + 490T + 2.26e5T^{2} \)
67 \( 1 - 812iT - 3.00e5T^{2} \)
71 \( 1 - 120T + 3.57e5T^{2} \)
73 \( 1 + 746iT - 3.89e5T^{2} \)
79 \( 1 + 152T + 4.93e5T^{2} \)
83 \( 1 - 804iT - 5.71e5T^{2} \)
89 \( 1 - 678T + 7.04e5T^{2} \)
97 \( 1 - 194iT - 9.12e5T^{2} \)
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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

−12.62930005829372530495542724517, −11.84566126881147538660492756759, −10.93556765869403135531574427146, −9.613172678273575213380080315908, −8.670232017103985326615636378902, −7.68355861699815976034773466784, −5.97953053592003115708021121908, −5.06377722316938025968189036615, −2.98999304149075140160991469129, −1.94805511515521008373367675398, 0.38565541424593073791654779510, 3.29529185605881722530371142892, 4.63354267210361173861952079542, 5.63854425080443732257268933434, 7.36434527055723895621211508998, 7.77435807833639952018293444663, 9.341902598557670757364025634196, 10.39420403241401402880704709064, 10.91310889903058461504184134554, 12.70074530777439346284895072078

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