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Properties

Label 2-150-5.4-c1-0-0
Degree $2$
Conductor $150$
Sign $-0.447 - 0.894i$
Analytic cond. $1.19775$
Root an. cond. $1.09442$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + i·2-s + i·3-s − 4-s − 6-s + 4i·7-s − i·8-s − 9-s − i·12-s + 2i·13-s − 4·14-s + 16-s − 6i·17-s − i·18-s + 4·19-s − 4·21-s + ⋯
L(s)  = 1  + 0.707i·2-s + 0.577i·3-s − 0.5·4-s − 0.408·6-s + 1.51i·7-s − 0.353i·8-s − 0.333·9-s − 0.288i·12-s + 0.554i·13-s − 1.06·14-s + 0.250·16-s − 1.45i·17-s − 0.235i·18-s + 0.917·19-s − 0.872·21-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(150\)    =    \(2 \cdot 3 \cdot 5^{2}\)
Sign: $-0.447 - 0.894i$
Analytic conductor: \(1.19775\)
Root analytic conductor: \(1.09442\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{150} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 150,\ (\ :1/2),\ -0.447 - 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.544708 + 0.881357i\)
\(L(\frac12)\) \(\approx\) \(0.544708 + 0.881357i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 - iT \)
5 \( 1 \)
good7 \( 1 - 4iT - 7T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 - 2iT - 13T^{2} \)
17 \( 1 + 6iT - 17T^{2} \)
19 \( 1 - 4T + 19T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 - 8T + 31T^{2} \)
37 \( 1 + 2iT - 37T^{2} \)
41 \( 1 + 6T + 41T^{2} \)
43 \( 1 + 4iT - 43T^{2} \)
47 \( 1 - 47T^{2} \)
53 \( 1 + 6iT - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + 10T + 61T^{2} \)
67 \( 1 - 4iT - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 2iT - 73T^{2} \)
79 \( 1 + 8T + 79T^{2} \)
83 \( 1 - 12iT - 83T^{2} \)
89 \( 1 + 18T + 89T^{2} \)
97 \( 1 + 2iT - 97T^{2} \)
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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

−13.65903418781624892598801003079, −12.16301303983113287659365630302, −11.57232332102479816450987379595, −9.941819242818724229118457896505, −9.156664271354851304940002344415, −8.303150793602142243576863666858, −6.85200004047624573112690645068, −5.61952237968355492352832454031, −4.72670202858721634233331620829, −2.87232562339919378192796826883, 1.19301600866377439702955307792, 3.23242613233288142627718190811, 4.56480584669636693327632140720, 6.24917907500364745105060801431, 7.52893777749673750646095101981, 8.428246467858918435959153272298, 10.02114576910288854614874100657, 10.59711490453538236750892836991, 11.72862706244732673515655909624, 12.73048483344986556804976084426

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