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Properties

Label 2-150-15.2-c7-0-24
Degree $2$
Conductor $150$
Sign $0.245 + 0.969i$
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 + (−0.750 − 46.7i)3-s − 64.0i·4-s + (268. + 260. i)6-s + (661. + 661. i)7-s + (362. + 362. i)8-s + (−2.18e3 + 70.1i)9-s − 6.67e3i·11-s + (−2.99e3 + 48.0i)12-s + (4.25e3 − 4.25e3i)13-s − 7.48e3·14-s − 4.09e3·16-s + (1.15e4 − 1.15e4i)17-s + (1.19e4 − 1.27e4i)18-s + 5.65e4i·19-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s + (−0.0160 − 0.999i)3-s − 0.500i·4-s + (0.507 + 0.491i)6-s + (0.729 + 0.729i)7-s + (0.250 + 0.250i)8-s + (−0.999 + 0.0320i)9-s − 1.51i·11-s + (−0.499 + 0.00801i)12-s + (0.536 − 0.536i)13-s − 0.729·14-s − 0.250·16-s + (0.568 − 0.568i)17-s + (0.483 − 0.515i)18-s + 1.89i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.245 + 0.969i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.245 + 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(150\)    =    \(2 \cdot 3 \cdot 5^{2}\)
Sign: $0.245 + 0.969i$
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: \((2,\ 150,\ (\ :7/2),\ 0.245 + 0.969i)\)

Particular Values

\(L(4)\) \(\approx\) \(1.549285946\)
\(L(\frac12)\) \(\approx\) \(1.549285946\)
\(L(\frac{9}{2})\) 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 \( 1 + (0.750 + 46.7i)T \)
5 \( 1 \)
good7 \( 1 + (-661. - 661. i)T + 8.23e5iT^{2} \)
11 \( 1 + 6.67e3iT - 1.94e7T^{2} \)
13 \( 1 + (-4.25e3 + 4.25e3i)T - 6.27e7iT^{2} \)
17 \( 1 + (-1.15e4 + 1.15e4i)T - 4.10e8iT^{2} \)
19 \( 1 - 5.65e4iT - 8.93e8T^{2} \)
23 \( 1 + (-3.68e4 - 3.68e4i)T + 3.40e9iT^{2} \)
29 \( 1 + 3.43e4T + 1.72e10T^{2} \)
31 \( 1 - 1.54e5T + 2.75e10T^{2} \)
37 \( 1 + (8.63e4 + 8.63e4i)T + 9.49e10iT^{2} \)
41 \( 1 + 5.20e5iT - 1.94e11T^{2} \)
43 \( 1 + (-4.99e5 + 4.99e5i)T - 2.71e11iT^{2} \)
47 \( 1 + (-4.71e4 + 4.71e4i)T - 5.06e11iT^{2} \)
53 \( 1 + (-3.15e5 - 3.15e5i)T + 1.17e12iT^{2} \)
59 \( 1 + 1.04e6T + 2.48e12T^{2} \)
61 \( 1 + 1.20e6T + 3.14e12T^{2} \)
67 \( 1 + (2.42e5 + 2.42e5i)T + 6.06e12iT^{2} \)
71 \( 1 - 3.41e4iT - 9.09e12T^{2} \)
73 \( 1 + (-2.31e6 + 2.31e6i)T - 1.10e13iT^{2} \)
79 \( 1 + 7.82e6iT - 1.92e13T^{2} \)
83 \( 1 + (5.24e6 + 5.24e6i)T + 2.71e13iT^{2} \)
89 \( 1 + 5.17e6T + 4.42e13T^{2} \)
97 \( 1 + (1.14e5 + 1.14e5i)T + 8.07e13iT^{2} \)
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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

−11.52638420605878934528846721308, −10.55616746471011188494186071467, −8.935737582925310153412544214585, −8.262478201117426844660310286188, −7.47593409009046536804969387808, −5.93527514044533818341003350510, −5.55681491514889901975985187630, −3.22569408344256229939944122733, −1.66974650596095551923432082222, −0.59208310091006467160891965008, 1.11750271827958720691982010511, 2.65110813785814079154078554669, 4.20105228012598375199797384763, 4.82128532744342991870341888683, 6.74503483332429036310347206918, 7.962962844353988178905431578491, 9.079072094313401146570215503282, 9.917971612307814084834245401037, 10.84971524214486008778665797828, 11.47488889942030088558546180632

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