Properties

Label 2-150-5.4-c15-0-36
Degree $2$
Conductor $150$
Sign $-0.447 + 0.894i$
Analytic cond. $214.040$
Root an. cond. $14.6301$
Motivic weight $15$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 128i·2-s + 2.18e3i·3-s − 1.63e4·4-s + 2.79e5·6-s + 2.02e6i·7-s + 2.09e6i·8-s − 4.78e6·9-s + 1.10e8·11-s − 3.58e7i·12-s − 5.60e7i·13-s + 2.59e8·14-s + 2.68e8·16-s − 1.93e9i·17-s + 6.12e8i·18-s − 2.16e9·19-s + ⋯
L(s)  = 1  − 0.707i·2-s + 0.577i·3-s − 0.5·4-s + 0.408·6-s + 0.929i·7-s + 0.353i·8-s − 0.333·9-s + 1.70·11-s − 0.288i·12-s − 0.247i·13-s + 0.657·14-s + 0.250·16-s − 1.14i·17-s + 0.235i·18-s − 0.555·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(150\)    =    \(2 \cdot 3 \cdot 5^{2}\)
Sign: $-0.447 + 0.894i$
Analytic conductor: \(214.040\)
Root analytic conductor: \(14.6301\)
Motivic weight: \(15\)
Rational: no
Arithmetic: yes
Character: $\chi_{150} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 150,\ (\ :15/2),\ -0.447 + 0.894i)\)

Particular Values

\(L(8)\) \(\approx\) \(1.251812412\)
\(L(\frac12)\) \(\approx\) \(1.251812412\)
\(L(\frac{17}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 128iT \)
3 \( 1 - 2.18e3iT \)
5 \( 1 \)
good7 \( 1 - 2.02e6iT - 4.74e12T^{2} \)
11 \( 1 - 1.10e8T + 4.17e15T^{2} \)
13 \( 1 + 5.60e7iT - 5.11e16T^{2} \)
17 \( 1 + 1.93e9iT - 2.86e18T^{2} \)
19 \( 1 + 2.16e9T + 1.51e19T^{2} \)
23 \( 1 + 6.22e9iT - 2.66e20T^{2} \)
29 \( 1 + 6.47e10T + 8.62e21T^{2} \)
31 \( 1 + 2.02e10T + 2.34e22T^{2} \)
37 \( 1 - 4.88e11iT - 3.33e23T^{2} \)
41 \( 1 + 7.72e11T + 1.55e24T^{2} \)
43 \( 1 + 1.30e12iT - 3.17e24T^{2} \)
47 \( 1 - 3.35e12iT - 1.20e25T^{2} \)
53 \( 1 + 9.38e12iT - 7.31e25T^{2} \)
59 \( 1 + 2.89e13T + 3.65e26T^{2} \)
61 \( 1 - 4.23e13T + 6.02e26T^{2} \)
67 \( 1 + 5.22e13iT - 2.46e27T^{2} \)
71 \( 1 + 2.71e13T + 5.87e27T^{2} \)
73 \( 1 - 9.16e13iT - 8.90e27T^{2} \)
79 \( 1 + 6.28e13T + 2.91e28T^{2} \)
83 \( 1 - 2.23e14iT - 6.11e28T^{2} \)
89 \( 1 + 5.54e14T + 1.74e29T^{2} \)
97 \( 1 + 1.38e15iT - 6.33e29T^{2} \)
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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

−9.850456436031321667198067893041, −9.179928138623861808774227493546, −8.427057722853394958732579445335, −6.79623885962521100326414517829, −5.64779575969216563462644011651, −4.59250185638389019798496076953, −3.60746665128186669238117183251, −2.59610809381468700034527407906, −1.49681853983695539882816281893, −0.24711664726830165566779003831, 0.975137189722167754304305341121, 1.78459645663232510525725483571, 3.63706403460923494927883511134, 4.29055384525819367591258564874, 5.84018814959322927136573761808, 6.66113043273219575097647249751, 7.38505025773387376626222836438, 8.489678576537484271571169311386, 9.373505983836601583491115634189, 10.59292533066996181437252268857

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