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Properties

Label 2-150-75.23-c1-0-6
Degree $2$
Conductor $150$
Sign $0.465 + 0.884i$
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  + (−0.987 − 0.156i)2-s + (0.487 − 1.66i)3-s + (0.951 + 0.309i)4-s + (2.22 − 0.193i)5-s + (−0.741 + 1.56i)6-s + (−0.104 + 0.104i)7-s + (−0.891 − 0.453i)8-s + (−2.52 − 1.62i)9-s + (−2.23 − 0.157i)10-s + (1.81 + 2.49i)11-s + (0.977 − 1.42i)12-s + (−0.711 − 4.49i)13-s + (0.119 − 0.0869i)14-s + (0.765 − 3.79i)15-s + (0.809 + 0.587i)16-s + (1.68 − 3.30i)17-s + ⋯
L(s)  = 1  + (−0.698 − 0.110i)2-s + (0.281 − 0.959i)3-s + (0.475 + 0.154i)4-s + (0.996 − 0.0863i)5-s + (−0.302 + 0.638i)6-s + (−0.0395 + 0.0395i)7-s + (−0.315 − 0.160i)8-s + (−0.841 − 0.540i)9-s + (−0.705 − 0.0499i)10-s + (0.546 + 0.752i)11-s + (0.282 − 0.412i)12-s + (−0.197 − 1.24i)13-s + (0.0319 − 0.0232i)14-s + (0.197 − 0.980i)15-s + (0.202 + 0.146i)16-s + (0.408 − 0.801i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.465 + 0.884i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.465 + 0.884i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(150\)    =    \(2 \cdot 3 \cdot 5^{2}\)
Sign: $0.465 + 0.884i$
Analytic conductor: \(1.19775\)
Root analytic conductor: \(1.09442\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{150} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 150,\ (\ :1/2),\ 0.465 + 0.884i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.862968 - 0.521092i\)
\(L(\frac12)\) \(\approx\) \(0.862968 - 0.521092i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.987 + 0.156i)T \)
3 \( 1 + (-0.487 + 1.66i)T \)
5 \( 1 + (-2.22 + 0.193i)T \)
good7 \( 1 + (0.104 - 0.104i)T - 7iT^{2} \)
11 \( 1 + (-1.81 - 2.49i)T + (-3.39 + 10.4i)T^{2} \)
13 \( 1 + (0.711 + 4.49i)T + (-12.3 + 4.01i)T^{2} \)
17 \( 1 + (-1.68 + 3.30i)T + (-9.99 - 13.7i)T^{2} \)
19 \( 1 + (3.32 - 1.08i)T + (15.3 - 11.1i)T^{2} \)
23 \( 1 + (0.773 - 4.88i)T + (-21.8 - 7.10i)T^{2} \)
29 \( 1 + (0.509 - 1.56i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-2.45 - 7.56i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (8.30 - 1.31i)T + (35.1 - 11.4i)T^{2} \)
41 \( 1 + (1.27 - 1.75i)T + (-12.6 - 38.9i)T^{2} \)
43 \( 1 + (-2.77 - 2.77i)T + 43iT^{2} \)
47 \( 1 + (-4.77 + 2.43i)T + (27.6 - 38.0i)T^{2} \)
53 \( 1 + (-5.24 - 10.2i)T + (-31.1 + 42.8i)T^{2} \)
59 \( 1 + (10.4 + 7.57i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-11.8 + 8.58i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (-3.23 - 1.64i)T + (39.3 + 54.2i)T^{2} \)
71 \( 1 + (4.79 + 1.55i)T + (57.4 + 41.7i)T^{2} \)
73 \( 1 + (10.6 + 1.68i)T + (69.4 + 22.5i)T^{2} \)
79 \( 1 + (-8.57 - 2.78i)T + (63.9 + 46.4i)T^{2} \)
83 \( 1 + (9.01 + 4.59i)T + (48.7 + 67.1i)T^{2} \)
89 \( 1 + (-6.13 + 4.45i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (4.20 + 8.26i)T + (-57.0 + 78.4i)T^{2} \)
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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

−12.68237666640749392027992911240, −12.07986629932997149314752500304, −10.63352957398576593280002286501, −9.655674530011888972267383720697, −8.746888336719848188525350723543, −7.58839816401912071805948503485, −6.63054683728749249068120754977, −5.44112315725967465731956123865, −2.93945904682430027384171912671, −1.50803087394075448672505676506, 2.26722932499397721999701850583, 4.06294334088641262266437646800, 5.69022772548555099244200485775, 6.69519132013679640796088341502, 8.464756265917050090055219144202, 9.091559558134752475793080783465, 10.06105618217741976638911495651, 10.78659299690367398630304297874, 11.89126224466003741982749524248, 13.46915881438020372275230545055

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