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Properties

Label 2-150-15.2-c7-0-18
Degree $2$
Conductor $150$
Sign $-0.854 - 0.520i$
Analytic cond. $46.8577$
Root an. cond. $6.84527$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−5.65 + 5.65i)2-s + (33.2 + 32.8i)3-s − 64.0i·4-s + (−374. + 2.40i)6-s + (919. + 919. i)7-s + (362. + 362. i)8-s + (28.1 + 2.18e3i)9-s + 7.17e3i·11-s + (2.10e3 − 2.12e3i)12-s + (1.91e3 − 1.91e3i)13-s − 1.04e4·14-s − 4.09e3·16-s + (1.51e4 − 1.51e4i)17-s + (−1.25e4 − 1.22e4i)18-s − 3.02e4i·19-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s + (0.711 + 0.702i)3-s − 0.500i·4-s + (−0.707 + 0.00455i)6-s + (1.01 + 1.01i)7-s + (0.250 + 0.250i)8-s + (0.0128 + 0.999i)9-s + 1.62i·11-s + (0.351 − 0.355i)12-s + (0.241 − 0.241i)13-s − 1.01·14-s − 0.250·16-s + (0.746 − 0.746i)17-s + (−0.506 − 0.493i)18-s − 1.01i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.854 - 0.520i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.854 - 0.520i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(150\)    =    \(2 \cdot 3 \cdot 5^{2}\)
Sign: $-0.854 - 0.520i$
Analytic conductor: \(46.8577\)
Root analytic conductor: \(6.84527\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{150} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 150,\ (\ :7/2),\ -0.854 - 0.520i)\)

Particular Values

\(L(4)\) \(\approx\) \(2.354826255\)
\(L(\frac12)\) \(\approx\) \(2.354826255\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (5.65 - 5.65i)T \)
3 \( 1 + (-33.2 - 32.8i)T \)
5 \( 1 \)
good7 \( 1 + (-919. - 919. i)T + 8.23e5iT^{2} \)
11 \( 1 - 7.17e3iT - 1.94e7T^{2} \)
13 \( 1 + (-1.91e3 + 1.91e3i)T - 6.27e7iT^{2} \)
17 \( 1 + (-1.51e4 + 1.51e4i)T - 4.10e8iT^{2} \)
19 \( 1 + 3.02e4iT - 8.93e8T^{2} \)
23 \( 1 + (-7.00e4 - 7.00e4i)T + 3.40e9iT^{2} \)
29 \( 1 - 6.41e4T + 1.72e10T^{2} \)
31 \( 1 + 8.21e4T + 2.75e10T^{2} \)
37 \( 1 + (-2.28e5 - 2.28e5i)T + 9.49e10iT^{2} \)
41 \( 1 - 6.18e4iT - 1.94e11T^{2} \)
43 \( 1 + (9.28e4 - 9.28e4i)T - 2.71e11iT^{2} \)
47 \( 1 + (-3.90e5 + 3.90e5i)T - 5.06e11iT^{2} \)
53 \( 1 + (4.52e5 + 4.52e5i)T + 1.17e12iT^{2} \)
59 \( 1 - 1.71e5T + 2.48e12T^{2} \)
61 \( 1 + 1.71e6T + 3.14e12T^{2} \)
67 \( 1 + (3.34e6 + 3.34e6i)T + 6.06e12iT^{2} \)
71 \( 1 + 2.46e5iT - 9.09e12T^{2} \)
73 \( 1 + (1.91e6 - 1.91e6i)T - 1.10e13iT^{2} \)
79 \( 1 - 1.41e6iT - 1.92e13T^{2} \)
83 \( 1 + (-4.78e6 - 4.78e6i)T + 2.71e13iT^{2} \)
89 \( 1 + 4.98e6T + 4.42e13T^{2} \)
97 \( 1 + (6.44e6 + 6.44e6i)T + 8.07e13iT^{2} \)
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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

−11.92102436051784068056651924704, −10.90878288046566208571063530281, −9.671433595217110383281810597881, −9.107144508899977924997199896474, −8.023366237053530496264230543696, −7.18277192543086085157498734659, −5.31806809065322294962032975767, −4.68857374417458825894773120348, −2.79404839921743637207696898654, −1.57809521294185030212162951117, 0.74710606433557688490152290383, 1.48058119290040039651804238150, 3.04728959412319056476711459913, 4.11124605270616022508307577840, 6.05571791603093820008711696854, 7.43507182290627455297786478096, 8.215560119432293543916582344684, 8.919768425781274257957238928386, 10.44218463406564805473978265607, 11.12151140444186297123969952435

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