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Properties

Label 2-150-75.53-c1-0-1
Degree $2$
Conductor $150$
Sign $-0.699 - 0.714i$
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  + (−0.891 + 0.453i)2-s + (−0.322 + 1.70i)3-s + (0.587 − 0.809i)4-s + (−0.0831 + 2.23i)5-s + (−0.485 − 1.66i)6-s + (0.0556 − 0.0556i)7-s + (−0.156 + 0.987i)8-s + (−2.79 − 1.09i)9-s + (−0.940 − 2.02i)10-s + (−1.04 − 0.340i)11-s + (1.18 + 1.26i)12-s + (−2.31 + 4.54i)13-s + (−0.0243 + 0.0748i)14-s + (−3.77 − 0.861i)15-s + (−0.309 − 0.951i)16-s + (3.10 + 0.491i)17-s + ⋯
L(s)  = 1  + (−0.630 + 0.321i)2-s + (−0.186 + 0.982i)3-s + (0.293 − 0.404i)4-s + (−0.0371 + 0.999i)5-s + (−0.198 − 0.678i)6-s + (0.0210 − 0.0210i)7-s + (−0.0553 + 0.349i)8-s + (−0.930 − 0.365i)9-s + (−0.297 − 0.641i)10-s + (−0.315 − 0.102i)11-s + (0.342 + 0.364i)12-s + (−0.641 + 1.25i)13-s + (−0.00649 + 0.0200i)14-s + (−0.974 − 0.222i)15-s + (−0.0772 − 0.237i)16-s + (0.752 + 0.119i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.699 - 0.714i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.699 - 0.714i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(150\)    =    \(2 \cdot 3 \cdot 5^{2}\)
Sign: $-0.699 - 0.714i$
Analytic conductor: \(1.19775\)
Root analytic conductor: \(1.09442\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{150} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 150,\ (\ :1/2),\ -0.699 - 0.714i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.269657 + 0.641732i\)
\(L(\frac12)\) \(\approx\) \(0.269657 + 0.641732i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.891 - 0.453i)T \)
3 \( 1 + (0.322 - 1.70i)T \)
5 \( 1 + (0.0831 - 2.23i)T \)
good7 \( 1 + (-0.0556 + 0.0556i)T - 7iT^{2} \)
11 \( 1 + (1.04 + 0.340i)T + (8.89 + 6.46i)T^{2} \)
13 \( 1 + (2.31 - 4.54i)T + (-7.64 - 10.5i)T^{2} \)
17 \( 1 + (-3.10 - 0.491i)T + (16.1 + 5.25i)T^{2} \)
19 \( 1 + (0.824 + 1.13i)T + (-5.87 + 18.0i)T^{2} \)
23 \( 1 + (-1.13 - 2.22i)T + (-13.5 + 18.6i)T^{2} \)
29 \( 1 + (-5.66 - 4.11i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-7.72 + 5.61i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-9.23 - 4.70i)T + (21.7 + 29.9i)T^{2} \)
41 \( 1 + (-3.55 + 1.15i)T + (33.1 - 24.0i)T^{2} \)
43 \( 1 + (1.00 + 1.00i)T + 43iT^{2} \)
47 \( 1 + (1.98 + 12.5i)T + (-44.6 + 14.5i)T^{2} \)
53 \( 1 + (6.70 - 1.06i)T + (50.4 - 16.3i)T^{2} \)
59 \( 1 + (-2.03 - 6.26i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-1.23 + 3.79i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (1.04 - 6.57i)T + (-63.7 - 20.7i)T^{2} \)
71 \( 1 + (7.51 - 10.3i)T + (-21.9 - 67.5i)T^{2} \)
73 \( 1 + (-1.18 + 0.602i)T + (42.9 - 59.0i)T^{2} \)
79 \( 1 + (-5.85 + 8.05i)T + (-24.4 - 75.1i)T^{2} \)
83 \( 1 + (0.137 - 0.871i)T + (-78.9 - 25.6i)T^{2} \)
89 \( 1 + (-1.01 + 3.11i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (10.2 - 1.62i)T + (92.2 - 29.9i)T^{2} \)
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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

−13.79807596108987755090890888966, −11.90102896494315344334862271241, −11.18783605813711700834153698987, −10.15425732365626212574693645266, −9.564718073487864982083422274189, −8.278819696073703186805943905657, −7.01669482648809182821527441367, −5.94955947085463841935864698689, −4.47579571289746162711663817432, −2.81488393560064313043482143393, 0.887741335939681819068339936386, 2.71671810405974884994271828035, 4.94469792095970612420716043400, 6.23099428388710134336528103501, 7.81760045807464976227036302199, 8.161568046784006854802113441678, 9.534721323742546718273289114631, 10.61134594379741979395064410437, 11.90365442531089614120176829752, 12.52083641953973657296437105808

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