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Properties

Label 2-150-15.2-c3-0-2
Degree $2$
Conductor $150$
Sign $-0.998 + 0.0618i$
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 + (−0.878 + 5.12i)3-s − 4.00i·4-s + (−6 − 8.48i)6-s + (18 + 18i)7-s + (5.65 + 5.65i)8-s + (−25.4 − 9i)9-s + 50.9i·11-s + (20.4 + 3.51i)12-s − 50.9·14-s − 16.0·16-s + (−59.3 + 59.3i)17-s + (48.7 − 23.2i)18-s − 124i·19-s + (−108. + 76.3i)21-s + (−72 − 72i)22-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s + (−0.169 + 0.985i)3-s − 0.500i·4-s + (−0.408 − 0.577i)6-s + (0.971 + 0.971i)7-s + (0.250 + 0.250i)8-s + (−0.942 − 0.333i)9-s + 1.39i·11-s + (0.492 + 0.0845i)12-s − 0.971·14-s − 0.250·16-s + (−0.847 + 0.847i)17-s + (0.638 − 0.304i)18-s − 1.49i·19-s + (−1.12 + 0.793i)21-s + (−0.697 − 0.697i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0618i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.998 + 0.0618i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(150\)    =    \(2 \cdot 3 \cdot 5^{2}\)
Sign: $-0.998 + 0.0618i$
Analytic conductor: \(8.85028\)
Root analytic conductor: \(2.97494\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{150} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 150,\ (\ :3/2),\ -0.998 + 0.0618i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0287039 - 0.927052i\)
\(L(\frac12)\) \(\approx\) \(0.0287039 - 0.927052i\)
\(L(\frac{5}{2})\) 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 \( 1 + (0.878 - 5.12i)T \)
5 \( 1 \)
good7 \( 1 + (-18 - 18i)T + 343iT^{2} \)
11 \( 1 - 50.9iT - 1.33e3T^{2} \)
13 \( 1 - 2.19e3iT^{2} \)
17 \( 1 + (59.3 - 59.3i)T - 4.91e3iT^{2} \)
19 \( 1 + 124iT - 6.85e3T^{2} \)
23 \( 1 + (29.6 + 29.6i)T + 1.21e4iT^{2} \)
29 \( 1 + 254.T + 2.43e4T^{2} \)
31 \( 1 - 88T + 2.97e4T^{2} \)
37 \( 1 + (-72 - 72i)T + 5.06e4iT^{2} \)
41 \( 1 + 50.9iT - 6.89e4T^{2} \)
43 \( 1 + (342 - 342i)T - 7.95e4iT^{2} \)
47 \( 1 + (318. - 318. i)T - 1.03e5iT^{2} \)
53 \( 1 + (-220. - 220. i)T + 1.48e5iT^{2} \)
59 \( 1 - 865.T + 2.05e5T^{2} \)
61 \( 1 - 434T + 2.26e5T^{2} \)
67 \( 1 + (18 + 18i)T + 3.00e5iT^{2} \)
71 \( 1 + 509. iT - 3.57e5T^{2} \)
73 \( 1 + (-360 + 360i)T - 3.89e5iT^{2} \)
79 \( 1 - 1.02e3iT - 4.93e5T^{2} \)
83 \( 1 + (-173. - 173. i)T + 5.71e5iT^{2} \)
89 \( 1 - 101.T + 7.04e5T^{2} \)
97 \( 1 + (-216 - 216i)T + 9.12e5iT^{2} \)
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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

−13.05885860388745847559244839270, −11.69549119299621575494085201548, −10.99048546665738467309024271867, −9.798302255457137765756553288251, −8.997146012212603341304007585373, −8.078666475792658080466047516179, −6.60391505896555994965186472373, −5.26117797842436516105067750178, −4.44622085957618714774452639525, −2.19605603205448084371999481877, 0.53345015637070189510536628929, 1.87675095617346923935682117083, 3.67435730702672811109486621017, 5.47614578783819193485956982658, 6.92722634701442242107813133448, 7.930997238022472426808396861984, 8.634674466776740521671367509096, 10.21241572223575557233235336799, 11.33359768601060381199486217939, 11.62194115143322421259711461877

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