Properties

Label 2-150-75.62-c1-0-7
Degree $2$
Conductor $150$
Sign $0.736 + 0.676i$
Analytic cond. $1.19775$
Root an. cond. $1.09442$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.987 − 0.156i)2-s + (0.0565 − 1.73i)3-s + (0.951 − 0.309i)4-s + (1.48 + 1.66i)5-s + (−0.214 − 1.71i)6-s + (−1.08 − 1.08i)7-s + (0.891 − 0.453i)8-s + (−2.99 − 0.195i)9-s + (1.73 + 1.41i)10-s + (−1.61 + 2.21i)11-s + (−0.481 − 1.66i)12-s + (0.355 − 2.24i)13-s + (−1.24 − 0.903i)14-s + (2.97 − 2.48i)15-s + (0.809 − 0.587i)16-s + (2.77 + 5.44i)17-s + ⋯
L(s)  = 1  + (0.698 − 0.110i)2-s + (0.0326 − 0.999i)3-s + (0.475 − 0.154i)4-s + (0.665 + 0.746i)5-s + (−0.0877 − 0.701i)6-s + (−0.410 − 0.410i)7-s + (0.315 − 0.160i)8-s + (−0.997 − 0.0653i)9-s + (0.547 + 0.447i)10-s + (−0.486 + 0.669i)11-s + (−0.138 − 0.480i)12-s + (0.0985 − 0.622i)13-s + (−0.332 − 0.241i)14-s + (0.767 − 0.641i)15-s + (0.202 − 0.146i)16-s + (0.673 + 1.32i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.736 + 0.676i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.736 + 0.676i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(150\)    =    \(2 \cdot 3 \cdot 5^{2}\)
Sign: $0.736 + 0.676i$
Analytic conductor: \(1.19775\)
Root analytic conductor: \(1.09442\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{150} (137, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 150,\ (\ :1/2),\ 0.736 + 0.676i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.53878 - 0.599636i\)
\(L(\frac12)\) \(\approx\) \(1.53878 - 0.599636i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.987 + 0.156i)T \)
3 \( 1 + (-0.0565 + 1.73i)T \)
5 \( 1 + (-1.48 - 1.66i)T \)
good7 \( 1 + (1.08 + 1.08i)T + 7iT^{2} \)
11 \( 1 + (1.61 - 2.21i)T + (-3.39 - 10.4i)T^{2} \)
13 \( 1 + (-0.355 + 2.24i)T + (-12.3 - 4.01i)T^{2} \)
17 \( 1 + (-2.77 - 5.44i)T + (-9.99 + 13.7i)T^{2} \)
19 \( 1 + (4.05 + 1.31i)T + (15.3 + 11.1i)T^{2} \)
23 \( 1 + (0.805 + 5.08i)T + (-21.8 + 7.10i)T^{2} \)
29 \( 1 + (-2.37 - 7.29i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (1.85 - 5.71i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (3.29 + 0.521i)T + (35.1 + 11.4i)T^{2} \)
41 \( 1 + (5.36 + 7.38i)T + (-12.6 + 38.9i)T^{2} \)
43 \( 1 + (-6.65 + 6.65i)T - 43iT^{2} \)
47 \( 1 + (-2.15 - 1.09i)T + (27.6 + 38.0i)T^{2} \)
53 \( 1 + (-0.199 + 0.391i)T + (-31.1 - 42.8i)T^{2} \)
59 \( 1 + (-5.86 + 4.26i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (8.14 + 5.91i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + (-5.28 + 2.69i)T + (39.3 - 54.2i)T^{2} \)
71 \( 1 + (5.89 - 1.91i)T + (57.4 - 41.7i)T^{2} \)
73 \( 1 + (-10.2 + 1.62i)T + (69.4 - 22.5i)T^{2} \)
79 \( 1 + (-6.06 + 1.97i)T + (63.9 - 46.4i)T^{2} \)
83 \( 1 + (6.92 - 3.52i)T + (48.7 - 67.1i)T^{2} \)
89 \( 1 + (8.12 + 5.90i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (-5.51 + 10.8i)T + (-57.0 - 78.4i)T^{2} \)
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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

−12.66623431900273693574894987718, −12.54203760721863677547183627306, −10.72933474886312101053405921466, −10.36032668383931714698006270223, −8.550246524411396943313348505523, −7.19895136074875159110147000308, −6.48893133163980359591370915373, −5.39034717897037561498767180784, −3.39690811089426487732510336894, −2.04730711244208248663716876959, 2.71175555699623959022338551693, 4.23157133730680949644448647605, 5.39305553813788925865544234080, 6.13877060818899774196333685312, 8.048579789289619341350114297945, 9.234216240543220211262076510219, 9.949824916641642350824766350524, 11.27105844921607857728026243167, 12.13633407028448576477755569168, 13.39778635094597024147432920806

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