Properties

Label 2-150-25.14-c1-0-1
Degree $2$
Conductor $150$
Sign $0.331 - 0.943i$
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  + (0.587 + 0.809i)2-s + (−0.951 − 0.309i)3-s + (−0.309 + 0.951i)4-s + (1.70 + 1.44i)5-s + (−0.309 − 0.951i)6-s + 1.72i·7-s + (−0.951 + 0.309i)8-s + (0.809 + 0.587i)9-s + (−0.166 + 2.22i)10-s + (1.97 − 1.43i)11-s + (0.587 − 0.809i)12-s + (−1.74 + 2.40i)13-s + (−1.39 + 1.01i)14-s + (−1.17 − 1.90i)15-s + (−0.809 − 0.587i)16-s + (3.18 − 1.03i)17-s + ⋯
L(s)  = 1  + (0.415 + 0.572i)2-s + (−0.549 − 0.178i)3-s + (−0.154 + 0.475i)4-s + (0.762 + 0.646i)5-s + (−0.126 − 0.388i)6-s + 0.652i·7-s + (−0.336 + 0.109i)8-s + (0.269 + 0.195i)9-s + (−0.0528 + 0.705i)10-s + (0.595 − 0.432i)11-s + (0.169 − 0.233i)12-s + (−0.484 + 0.666i)13-s + (−0.373 + 0.271i)14-s + (−0.303 − 0.491i)15-s + (−0.202 − 0.146i)16-s + (0.771 − 0.250i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.331 - 0.943i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.331 - 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(150\)    =    \(2 \cdot 3 \cdot 5^{2}\)
Sign: $0.331 - 0.943i$
Analytic conductor: \(1.19775\)
Root analytic conductor: \(1.09442\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{150} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 150,\ (\ :1/2),\ 0.331 - 0.943i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.03185 + 0.731064i\)
\(L(\frac12)\) \(\approx\) \(1.03185 + 0.731064i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.587 - 0.809i)T \)
3 \( 1 + (0.951 + 0.309i)T \)
5 \( 1 + (-1.70 - 1.44i)T \)
good7 \( 1 - 1.72iT - 7T^{2} \)
11 \( 1 + (-1.97 + 1.43i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (1.74 - 2.40i)T + (-4.01 - 12.3i)T^{2} \)
17 \( 1 + (-3.18 + 1.03i)T + (13.7 - 9.99i)T^{2} \)
19 \( 1 + (0.694 + 2.13i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (5.35 + 7.37i)T + (-7.10 + 21.8i)T^{2} \)
29 \( 1 + (-2.89 + 8.91i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (1.89 + 5.82i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (1.94 - 2.67i)T + (-11.4 - 35.1i)T^{2} \)
41 \( 1 + (-4.95 - 3.60i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 7.06iT - 43T^{2} \)
47 \( 1 + (-4.74 - 1.54i)T + (38.0 + 27.6i)T^{2} \)
53 \( 1 + (6.51 + 2.11i)T + (42.8 + 31.1i)T^{2} \)
59 \( 1 + (0.147 + 0.107i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (4.16 - 3.02i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (11.3 - 3.67i)T + (54.2 - 39.3i)T^{2} \)
71 \( 1 + (3.51 - 10.8i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (3.95 + 5.44i)T + (-22.5 + 69.4i)T^{2} \)
79 \( 1 + (-3.50 + 10.8i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-8.69 + 2.82i)T + (67.1 - 48.7i)T^{2} \)
89 \( 1 + (6.56 - 4.77i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (-9.40 - 3.05i)T + (78.4 + 57.0i)T^{2} \)
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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

−13.36401363319109056136939421980, −12.19082214958644767819682936900, −11.47914732779756427017271812649, −10.13557795265158226389225525200, −9.117485292857677072054974549902, −7.70996543172924981791598625798, −6.39195372657969782857804921467, −5.93704100825626943491352389076, −4.45537292895232149990941377331, −2.52240532732498267869034480613, 1.49307621749145247293830725356, 3.70197360395045360126036730490, 5.03272960524734459525352367324, 5.91963445896214015498339546110, 7.38969321860464165292409945248, 9.044824841764921234926967135605, 10.06861811261560574738786995926, 10.64128525442051248842969806563, 12.18714908668785071595874203233, 12.48823334027508967425485257994

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