Properties

Label 2-150-5.3-c2-0-4
Degree $2$
Conductor $150$
Sign $-0.850 + 0.525i$
Analytic cond. $4.08720$
Root an. cond. $2.02168$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1 + i)2-s + (−1.22 − 1.22i)3-s − 2i·4-s + 2.44·6-s + (−0.898 + 0.898i)7-s + (2 + 2i)8-s + 2.99i·9-s − 13.7·11-s + (−2.44 + 2.44i)12-s + (−12.7 − 12.7i)13-s − 1.79i·14-s − 4·16-s + (−15.8 + 15.8i)17-s + (−2.99 − 2.99i)18-s − 25.7i·19-s + ⋯
L(s)  = 1  + (−0.5 + 0.5i)2-s + (−0.408 − 0.408i)3-s − 0.5i·4-s + 0.408·6-s + (−0.128 + 0.128i)7-s + (0.250 + 0.250i)8-s + 0.333i·9-s − 1.25·11-s + (−0.204 + 0.204i)12-s + (−0.984 − 0.984i)13-s − 0.128i·14-s − 0.250·16-s + (−0.935 + 0.935i)17-s + (−0.166 − 0.166i)18-s − 1.35i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.850 + 0.525i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.850 + 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(150\)    =    \(2 \cdot 3 \cdot 5^{2}\)
Sign: $-0.850 + 0.525i$
Analytic conductor: \(4.08720\)
Root analytic conductor: \(2.02168\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{150} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 150,\ (\ :1),\ -0.850 + 0.525i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0483287 - 0.170124i\)
\(L(\frac12)\) \(\approx\) \(0.0483287 - 0.170124i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1 - i)T \)
3 \( 1 + (1.22 + 1.22i)T \)
5 \( 1 \)
good7 \( 1 + (0.898 - 0.898i)T - 49iT^{2} \)
11 \( 1 + 13.7T + 121T^{2} \)
13 \( 1 + (12.7 + 12.7i)T + 169iT^{2} \)
17 \( 1 + (15.8 - 15.8i)T - 289iT^{2} \)
19 \( 1 + 25.7iT - 361T^{2} \)
23 \( 1 + (10.6 + 10.6i)T + 529iT^{2} \)
29 \( 1 - 25.7iT - 841T^{2} \)
31 \( 1 + 39.5T + 961T^{2} \)
37 \( 1 + (-27 + 27i)T - 1.36e3iT^{2} \)
41 \( 1 - 17.7T + 1.68e3T^{2} \)
43 \( 1 + (-12.4 - 12.4i)T + 1.84e3iT^{2} \)
47 \( 1 + (9.30 - 9.30i)T - 2.20e3iT^{2} \)
53 \( 1 + (-19.0 - 19.0i)T + 2.80e3iT^{2} \)
59 \( 1 + 20iT - 3.48e3T^{2} \)
61 \( 1 - 15.1T + 3.72e3T^{2} \)
67 \( 1 + (-48.0 + 48.0i)T - 4.48e3iT^{2} \)
71 \( 1 - 6.20T + 5.04e3T^{2} \)
73 \( 1 + (-37.2 - 37.2i)T + 5.32e3iT^{2} \)
79 \( 1 + 115. iT - 6.24e3T^{2} \)
83 \( 1 + (82.2 + 82.2i)T + 6.88e3iT^{2} \)
89 \( 1 + 117. iT - 7.92e3T^{2} \)
97 \( 1 + (81.9 - 81.9i)T - 9.40e3iT^{2} \)
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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

−12.66041507798360231989396977534, −11.06210340680895822954254760922, −10.44301357574432506219193196522, −9.155079518718766339670725072935, −7.980092986599776365741395089227, −7.14342027454785715342519085476, −5.88835116358686602141676055990, −4.85287465499011537611036602765, −2.46916951884235372074449748693, −0.12932674921362904919394375637, 2.34636973686680268737516353970, 4.08437209567880076504784101286, 5.38035734775189934034884366767, 6.96493848515660313922851764773, 8.059914367064029189004579170825, 9.448853024049826765891812105246, 10.06893570799668144457871089459, 11.15444544549919937431452457772, 11.95047838264274956954805476474, 12.98110289934843233281428072140

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