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Properties

Label 2-150-15.2-c7-0-19
Degree $2$
Conductor $150$
Sign $0.748 + 0.663i$
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 + (−38.2 + 26.9i)3-s − 64.0i·4-s + (−63.8 + 368. i)6-s + (−6.12 − 6.12i)7-s + (−362. − 362. i)8-s + (735. − 2.05e3i)9-s + 2.86e3i·11-s + (1.72e3 + 2.44e3i)12-s + (−2.01e3 + 2.01e3i)13-s − 69.3·14-s − 4.09e3·16-s + (−8.71e3 + 8.71e3i)17-s + (−7.49e3 − 1.58e4i)18-s − 1.40e3i·19-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s + (−0.817 + 0.576i)3-s − 0.500i·4-s + (−0.120 + 0.696i)6-s + (−0.00675 − 0.00675i)7-s + (−0.250 − 0.250i)8-s + (0.336 − 0.941i)9-s + 0.648i·11-s + (0.288 + 0.408i)12-s + (−0.254 + 0.254i)13-s − 0.00675·14-s − 0.250·16-s + (−0.430 + 0.430i)17-s + (−0.302 − 0.639i)18-s − 0.0471i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.748 + 0.663i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.748 + 0.663i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(150\)    =    \(2 \cdot 3 \cdot 5^{2}\)
Sign: $0.748 + 0.663i$
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: \((2,\ 150,\ (\ :7/2),\ 0.748 + 0.663i)\)

Particular Values

\(L(4)\) \(\approx\) \(1.734891955\)
\(L(\frac12)\) \(\approx\) \(1.734891955\)
\(L(\frac{9}{2})\) 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 \( 1 + (38.2 - 26.9i)T \)
5 \( 1 \)
good7 \( 1 + (6.12 + 6.12i)T + 8.23e5iT^{2} \)
11 \( 1 - 2.86e3iT - 1.94e7T^{2} \)
13 \( 1 + (2.01e3 - 2.01e3i)T - 6.27e7iT^{2} \)
17 \( 1 + (8.71e3 - 8.71e3i)T - 4.10e8iT^{2} \)
19 \( 1 + 1.40e3iT - 8.93e8T^{2} \)
23 \( 1 + (5.24e4 + 5.24e4i)T + 3.40e9iT^{2} \)
29 \( 1 - 9.58e4T + 1.72e10T^{2} \)
31 \( 1 - 1.25e5T + 2.75e10T^{2} \)
37 \( 1 + (-3.05e5 - 3.05e5i)T + 9.49e10iT^{2} \)
41 \( 1 + 6.09e5iT - 1.94e11T^{2} \)
43 \( 1 + (-3.17e5 + 3.17e5i)T - 2.71e11iT^{2} \)
47 \( 1 + (-4.35e5 + 4.35e5i)T - 5.06e11iT^{2} \)
53 \( 1 + (-1.07e6 - 1.07e6i)T + 1.17e12iT^{2} \)
59 \( 1 - 3.09e6T + 2.48e12T^{2} \)
61 \( 1 - 1.62e6T + 3.14e12T^{2} \)
67 \( 1 + (9.07e5 + 9.07e5i)T + 6.06e12iT^{2} \)
71 \( 1 + 3.60e6iT - 9.09e12T^{2} \)
73 \( 1 + (2.84e6 - 2.84e6i)T - 1.10e13iT^{2} \)
79 \( 1 + 4.14e6iT - 1.92e13T^{2} \)
83 \( 1 + (2.77e6 + 2.77e6i)T + 2.71e13iT^{2} \)
89 \( 1 - 1.02e7T + 4.42e13T^{2} \)
97 \( 1 + (-2.22e6 - 2.22e6i)T + 8.07e13iT^{2} \)
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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

−11.75293645922753890698686786615, −10.51295925303438223309696427330, −9.994107175821207940169412941934, −8.721719466378078161458722557384, −6.96650396153958170518761776744, −5.95002876317506557430244380683, −4.74634901432683935268521569946, −3.94664815632819468127401970909, −2.27089685178641427114501421654, −0.62716331060160982821928321731, 0.829473304388402697881820578639, 2.57167691343378046309250916080, 4.26964439191734611822218463431, 5.48064755253146867873945573981, 6.29859875087767801105748409451, 7.39024921722491480047338229621, 8.320559992285198471004966510449, 9.852740644062403149430343305546, 11.15682463573785630025036262667, 11.85606223255835777977656076815

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