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Properties

Label 2-150-5.4-c3-0-0
Degree $2$
Conductor $150$
Sign $-0.447 - 0.894i$
Analytic cond. $8.85028$
Root an. cond. $2.97494$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2i·2-s + 3i·3-s − 4·4-s + 6·6-s + 4i·7-s + 8i·8-s − 9·9-s − 48·11-s − 12i·12-s + 2i·13-s + 8·14-s + 16·16-s + 114i·17-s + 18i·18-s − 140·19-s + ⋯
L(s)  = 1  − 0.707i·2-s + 0.577i·3-s − 0.5·4-s + 0.408·6-s + 0.215i·7-s + 0.353i·8-s − 0.333·9-s − 1.31·11-s − 0.288i·12-s + 0.0426i·13-s + 0.152·14-s + 0.250·16-s + 1.62i·17-s + 0.235i·18-s − 1.69·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(150\)    =    \(2 \cdot 3 \cdot 5^{2}\)
Sign: $-0.447 - 0.894i$
Analytic conductor: \(8.85028\)
Root analytic conductor: \(2.97494\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{150} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 150,\ (\ :3/2),\ -0.447 - 0.894i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.311319 + 0.503725i\)
\(L(\frac12)\) \(\approx\) \(0.311319 + 0.503725i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 2iT \)
3 \( 1 - 3iT \)
5 \( 1 \)
good7 \( 1 - 4iT - 343T^{2} \)
11 \( 1 + 48T + 1.33e3T^{2} \)
13 \( 1 - 2iT - 2.19e3T^{2} \)
17 \( 1 - 114iT - 4.91e3T^{2} \)
19 \( 1 + 140T + 6.85e3T^{2} \)
23 \( 1 - 72iT - 1.21e4T^{2} \)
29 \( 1 + 210T + 2.43e4T^{2} \)
31 \( 1 - 272T + 2.97e4T^{2} \)
37 \( 1 - 334iT - 5.06e4T^{2} \)
41 \( 1 + 198T + 6.89e4T^{2} \)
43 \( 1 + 268iT - 7.95e4T^{2} \)
47 \( 1 + 216iT - 1.03e5T^{2} \)
53 \( 1 + 78iT - 1.48e5T^{2} \)
59 \( 1 + 240T + 2.05e5T^{2} \)
61 \( 1 - 302T + 2.26e5T^{2} \)
67 \( 1 + 596iT - 3.00e5T^{2} \)
71 \( 1 + 768T + 3.57e5T^{2} \)
73 \( 1 + 478iT - 3.89e5T^{2} \)
79 \( 1 - 640T + 4.93e5T^{2} \)
83 \( 1 + 348iT - 5.71e5T^{2} \)
89 \( 1 + 210T + 7.04e5T^{2} \)
97 \( 1 - 1.53e3iT - 9.12e5T^{2} \)
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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

−12.86979543112994399977079057624, −11.78821021563569664780997133751, −10.59147494896833370313373735651, −10.22576905109508936003417735427, −8.815830045508566178127693547766, −8.008723332242600693263282145555, −6.13617407268324372371099619580, −4.89803569275988450126733203883, −3.64809476994345000178108283069, −2.15063714118031055240266891461, 0.26970457103126915749408202189, 2.58079993469889873831825000244, 4.54470384811990768391320096075, 5.76280917360728651188354811322, 6.96407135356038098304008064783, 7.82624810855910269559865874379, 8.823280012398518663406337505394, 10.12795762288647170570320878774, 11.21083133443429788769480921893, 12.56129776480021685096843606329

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