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Properties

Label 2-150-75.62-c1-0-2
Degree $2$
Conductor $150$
Sign $0.999 + 0.0187i$
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 + (1.10 − 1.33i)3-s + (0.951 − 0.309i)4-s + (−0.334 + 2.21i)5-s + (−0.883 + 1.48i)6-s + (2.78 + 2.78i)7-s + (−0.891 + 0.453i)8-s + (−0.555 − 2.94i)9-s + (−0.0154 − 2.23i)10-s + (2.08 − 2.86i)11-s + (0.639 − 1.60i)12-s + (0.331 − 2.09i)13-s + (−3.18 − 2.31i)14-s + (2.57 + 2.89i)15-s + (0.809 − 0.587i)16-s + (0.476 + 0.934i)17-s + ⋯
L(s)  = 1  + (−0.698 + 0.110i)2-s + (0.638 − 0.769i)3-s + (0.475 − 0.154i)4-s + (−0.149 + 0.988i)5-s + (−0.360 + 0.608i)6-s + (1.05 + 1.05i)7-s + (−0.315 + 0.160i)8-s + (−0.185 − 0.982i)9-s + (−0.00488 − 0.707i)10-s + (0.627 − 0.863i)11-s + (0.184 − 0.464i)12-s + (0.0920 − 0.581i)13-s + (−0.850 − 0.617i)14-s + (0.665 + 0.746i)15-s + (0.202 − 0.146i)16-s + (0.115 + 0.226i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0187i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0187i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(150\)    =    \(2 \cdot 3 \cdot 5^{2}\)
Sign: $0.999 + 0.0187i$
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: \((2,\ 150,\ (\ :1/2),\ 0.999 + 0.0187i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.05759 - 0.00989541i\)
\(L(\frac12)\) \(\approx\) \(1.05759 - 0.00989541i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.10 + 1.33i)T \)
5 \( 1 + (0.334 - 2.21i)T \)
good7 \( 1 + (-2.78 - 2.78i)T + 7iT^{2} \)
11 \( 1 + (-2.08 + 2.86i)T + (-3.39 - 10.4i)T^{2} \)
13 \( 1 + (-0.331 + 2.09i)T + (-12.3 - 4.01i)T^{2} \)
17 \( 1 + (-0.476 - 0.934i)T + (-9.99 + 13.7i)T^{2} \)
19 \( 1 + (4.71 + 1.53i)T + (15.3 + 11.1i)T^{2} \)
23 \( 1 + (-1.22 - 7.70i)T + (-21.8 + 7.10i)T^{2} \)
29 \( 1 + (0.858 + 2.64i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-0.458 + 1.41i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (1.07 + 0.169i)T + (35.1 + 11.4i)T^{2} \)
41 \( 1 + (0.691 + 0.952i)T + (-12.6 + 38.9i)T^{2} \)
43 \( 1 + (8.25 - 8.25i)T - 43iT^{2} \)
47 \( 1 + (10.2 + 5.24i)T + (27.6 + 38.0i)T^{2} \)
53 \( 1 + (-1.93 + 3.79i)T + (-31.1 - 42.8i)T^{2} \)
59 \( 1 + (-5.11 + 3.71i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-1.09 - 0.794i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + (-8.29 + 4.22i)T + (39.3 - 54.2i)T^{2} \)
71 \( 1 + (2.47 - 0.805i)T + (57.4 - 41.7i)T^{2} \)
73 \( 1 + (-8.82 + 1.39i)T + (69.4 - 22.5i)T^{2} \)
79 \( 1 + (14.4 - 4.69i)T + (63.9 - 46.4i)T^{2} \)
83 \( 1 + (9.56 - 4.87i)T + (48.7 - 67.1i)T^{2} \)
89 \( 1 + (2.14 + 1.55i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (-7.23 + 14.2i)T + (-57.0 - 78.4i)T^{2} \)
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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

−13.05677705896105437029314566141, −11.59366591143000481951088961022, −11.31722599022734046122546291694, −9.758854605723463723455874924951, −8.533952307106673411261272636224, −8.038102057616968184821803326699, −6.79252678592208293933052838946, −5.77162953789869180494380643170, −3.30100439277539942332629390909, −1.92184061831760861864037204907, 1.75310100353151414023777619084, 4.07392391529349870100407676168, 4.80368424182635591722231546195, 6.98722897878635188652361669101, 8.197084744663276876092332212564, 8.790521964312260095289798739806, 9.907274463504935687492800880188, 10.75006716234893215893711624382, 11.81025182221571953148605172852, 12.99892199107940535296606179838

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