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Properties

Label 2-150-15.8-c3-0-15
Degree $2$
Conductor $150$
Sign $0.414 + 0.910i$
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  + (1.41 + 1.41i)2-s + (−0.489 − 5.17i)3-s + 4.00i·4-s + (6.62 − 8.00i)6-s + (16.6 − 16.6i)7-s + (−5.65 + 5.65i)8-s + (−26.5 + 5.06i)9-s − 36.5i·11-s + (20.6 − 1.95i)12-s + (−22.1 − 22.1i)13-s + 47.1·14-s − 16.0·16-s + (55.9 + 55.9i)17-s + (−44.6 − 30.3i)18-s − 132. i·19-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + (−0.0942 − 0.995i)3-s + 0.500i·4-s + (0.450 − 0.544i)6-s + (0.900 − 0.900i)7-s + (−0.250 + 0.250i)8-s + (−0.982 + 0.187i)9-s − 1.00i·11-s + (0.497 − 0.0471i)12-s + (−0.473 − 0.473i)13-s + 0.900·14-s − 0.250·16-s + (0.797 + 0.797i)17-s + (−0.584 − 0.397i)18-s − 1.59i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.414 + 0.910i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.414 + 0.910i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(150\)    =    \(2 \cdot 3 \cdot 5^{2}\)
Sign: $0.414 + 0.910i$
Analytic conductor: \(8.85028\)
Root analytic conductor: \(2.97494\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{150} (143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 150,\ (\ :3/2),\ 0.414 + 0.910i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.71629 - 1.10487i\)
\(L(\frac12)\) \(\approx\) \(1.71629 - 1.10487i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.41 - 1.41i)T \)
3 \( 1 + (0.489 + 5.17i)T \)
5 \( 1 \)
good7 \( 1 + (-16.6 + 16.6i)T - 343iT^{2} \)
11 \( 1 + 36.5iT - 1.33e3T^{2} \)
13 \( 1 + (22.1 + 22.1i)T + 2.19e3iT^{2} \)
17 \( 1 + (-55.9 - 55.9i)T + 4.91e3iT^{2} \)
19 \( 1 + 132. iT - 6.85e3T^{2} \)
23 \( 1 + (-126. + 126. i)T - 1.21e4iT^{2} \)
29 \( 1 - 16.4T + 2.43e4T^{2} \)
31 \( 1 + 225.T + 2.97e4T^{2} \)
37 \( 1 + (-57.8 + 57.8i)T - 5.06e4iT^{2} \)
41 \( 1 - 479. iT - 6.89e4T^{2} \)
43 \( 1 + (-115. - 115. i)T + 7.95e4iT^{2} \)
47 \( 1 + (-254. - 254. i)T + 1.03e5iT^{2} \)
53 \( 1 + (153. - 153. i)T - 1.48e5iT^{2} \)
59 \( 1 - 117.T + 2.05e5T^{2} \)
61 \( 1 - 182.T + 2.26e5T^{2} \)
67 \( 1 + (506. - 506. i)T - 3.00e5iT^{2} \)
71 \( 1 - 967. iT - 3.57e5T^{2} \)
73 \( 1 + (275. + 275. i)T + 3.89e5iT^{2} \)
79 \( 1 - 106. iT - 4.93e5T^{2} \)
83 \( 1 + (-884. + 884. i)T - 5.71e5iT^{2} \)
89 \( 1 + 126.T + 7.04e5T^{2} \)
97 \( 1 + (-932. + 932. i)T - 9.12e5iT^{2} \)
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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

−12.72994167087947812447077417755, −11.39208206391293925865587428917, −10.77605225399414577613495828678, −8.806343185121475373058213105859, −7.85783310843449720306282317314, −7.07861773101707578929653644513, −5.87986461387494303369083915641, −4.68746869655458655480475991092, −2.93040225214743876415461984285, −0.902950246556946616339573193837, 2.02957625190686262131733229140, 3.62826451094231886911873875229, 4.96996659550469699038948327077, 5.57847673740472446565740437252, 7.47929900130197750115192948695, 8.979863125131392981830529771485, 9.751459309060906240793923509363, 10.79811816679832785068428560770, 11.85800008838461649897601698294, 12.29567285578643442499067569046

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