Properties

Label 2-150-25.21-c5-0-10
Degree $2$
Conductor $150$
Sign $0.189 - 0.981i$
Analytic cond. $24.0575$
Root an. cond. $4.90485$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.23 + 3.80i)2-s + (−7.28 − 5.29i)3-s + (−12.9 − 9.40i)4-s + (43.0 + 35.6i)5-s + (29.1 − 21.1i)6-s + 32.5·7-s + (51.7 − 37.6i)8-s + (25.0 + 77.0i)9-s + (−188. + 119. i)10-s + (−95.0 + 292. i)11-s + (44.4 + 136. i)12-s + (−262. − 807. i)13-s + (−40.2 + 123. i)14-s + (−124. − 487. i)15-s + (79.1 + 243. i)16-s + (870. − 632. i)17-s + ⋯
L(s)  = 1  + (−0.218 + 0.672i)2-s + (−0.467 − 0.339i)3-s + (−0.404 − 0.293i)4-s + (0.769 + 0.638i)5-s + (0.330 − 0.239i)6-s + 0.251·7-s + (0.286 − 0.207i)8-s + (0.103 + 0.317i)9-s + (−0.597 + 0.378i)10-s + (−0.236 + 0.728i)11-s + (0.0892 + 0.274i)12-s + (−0.430 − 1.32i)13-s + (−0.0548 + 0.168i)14-s + (−0.142 − 0.559i)15-s + (0.0772 + 0.237i)16-s + (0.730 − 0.530i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.189 - 0.981i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.189 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(150\)    =    \(2 \cdot 3 \cdot 5^{2}\)
Sign: $0.189 - 0.981i$
Analytic conductor: \(24.0575\)
Root analytic conductor: \(4.90485\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{150} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 150,\ (\ :5/2),\ 0.189 - 0.981i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.565545601\)
\(L(\frac12)\) \(\approx\) \(1.565545601\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.23 - 3.80i)T \)
3 \( 1 + (7.28 + 5.29i)T \)
5 \( 1 + (-43.0 - 35.6i)T \)
good7 \( 1 - 32.5T + 1.68e4T^{2} \)
11 \( 1 + (95.0 - 292. i)T + (-1.30e5 - 9.46e4i)T^{2} \)
13 \( 1 + (262. + 807. i)T + (-3.00e5 + 2.18e5i)T^{2} \)
17 \( 1 + (-870. + 632. i)T + (4.38e5 - 1.35e6i)T^{2} \)
19 \( 1 + (-2.24e3 + 1.63e3i)T + (7.65e5 - 2.35e6i)T^{2} \)
23 \( 1 + (1.42e3 - 4.39e3i)T + (-5.20e6 - 3.78e6i)T^{2} \)
29 \( 1 + (-2.74e3 - 1.99e3i)T + (6.33e6 + 1.95e7i)T^{2} \)
31 \( 1 + (3.65e3 - 2.65e3i)T + (8.84e6 - 2.72e7i)T^{2} \)
37 \( 1 + (-1.12e3 - 3.44e3i)T + (-5.61e7 + 4.07e7i)T^{2} \)
41 \( 1 + (-2.02e3 - 6.22e3i)T + (-9.37e7 + 6.80e7i)T^{2} \)
43 \( 1 - 1.48e4T + 1.47e8T^{2} \)
47 \( 1 + (-2.00e4 - 1.45e4i)T + (7.08e7 + 2.18e8i)T^{2} \)
53 \( 1 + (2.76e3 + 2.00e3i)T + (1.29e8 + 3.97e8i)T^{2} \)
59 \( 1 + (-7.01e3 - 2.16e4i)T + (-5.78e8 + 4.20e8i)T^{2} \)
61 \( 1 + (1.47e4 - 4.54e4i)T + (-6.83e8 - 4.96e8i)T^{2} \)
67 \( 1 + (-5.36e4 + 3.89e4i)T + (4.17e8 - 1.28e9i)T^{2} \)
71 \( 1 + (1.10e4 + 8.03e3i)T + (5.57e8 + 1.71e9i)T^{2} \)
73 \( 1 + (2.72e4 - 8.38e4i)T + (-1.67e9 - 1.21e9i)T^{2} \)
79 \( 1 + (-5.82e4 - 4.23e4i)T + (9.50e8 + 2.92e9i)T^{2} \)
83 \( 1 + (-5.91e4 + 4.29e4i)T + (1.21e9 - 3.74e9i)T^{2} \)
89 \( 1 + (-2.41e4 + 7.44e4i)T + (-4.51e9 - 3.28e9i)T^{2} \)
97 \( 1 + (7.53e4 + 5.47e4i)T + (2.65e9 + 8.16e9i)T^{2} \)
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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

−12.43179611483775742788458634660, −11.20924781398781789459243099137, −10.11263091958993595836256239520, −9.418691987656438548798062661184, −7.64461422796854915632289417785, −7.21022197599074792438707748824, −5.73928172153462619066330053366, −5.11947856017432369171706931111, −2.90073879268713424021941515604, −1.13914129632322709672222692167, 0.73625743007776968830816974114, 2.11364527697505680212847551403, 3.92040328002773324761003661268, 5.14106266342588512929814509267, 6.19055172562457281624067487848, 7.943631735791446591838324430351, 9.084822478349124694851781351449, 9.899245473720770900132163551398, 10.78725639492729363093423274201, 11.96234951137711226131872476160

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