Properties

Label 2-75-3.2-c8-0-9
Degree $2$
Conductor $75$
Sign $0.988 + 0.152i$
Analytic cond. $30.5533$
Root an. cond. $5.52751$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 29.2i·2-s + (80.0 + 12.3i)3-s − 596.·4-s + (360. − 2.33e3i)6-s − 2.07e3·7-s + 9.94e3i·8-s + (6.25e3 + 1.97e3i)9-s + 1.19e4i·11-s + (−4.77e4 − 7.37e3i)12-s − 2.78e4·13-s + 6.05e4i·14-s + 1.37e5·16-s + 3.24e4i·17-s + (5.77e4 − 1.82e5i)18-s + 1.22e5·19-s + ⋯
L(s)  = 1  − 1.82i·2-s + (0.988 + 0.152i)3-s − 2.33·4-s + (0.278 − 1.80i)6-s − 0.863·7-s + 2.42i·8-s + (0.953 + 0.301i)9-s + 0.817i·11-s + (−2.30 − 0.355i)12-s − 0.973·13-s + 1.57i·14-s + 2.10·16-s + 0.388i·17-s + (0.550 − 1.74i)18-s + 0.943·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 + 0.152i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.988 + 0.152i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(75\)    =    \(3 \cdot 5^{2}\)
Sign: $0.988 + 0.152i$
Analytic conductor: \(30.5533\)
Root analytic conductor: \(5.52751\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{75} (26, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 75,\ (\ :4),\ 0.988 + 0.152i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(1.34758 - 0.103417i\)
\(L(\frac12)\) \(\approx\) \(1.34758 - 0.103417i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-80.0 - 12.3i)T \)
5 \( 1 \)
good2 \( 1 + 29.2iT - 256T^{2} \)
7 \( 1 + 2.07e3T + 5.76e6T^{2} \)
11 \( 1 - 1.19e4iT - 2.14e8T^{2} \)
13 \( 1 + 2.78e4T + 8.15e8T^{2} \)
17 \( 1 - 3.24e4iT - 6.97e9T^{2} \)
19 \( 1 - 1.22e5T + 1.69e10T^{2} \)
23 \( 1 - 2.66e5iT - 7.83e10T^{2} \)
29 \( 1 - 7.93e5iT - 5.00e11T^{2} \)
31 \( 1 - 8.76e5T + 8.52e11T^{2} \)
37 \( 1 + 1.82e6T + 3.51e12T^{2} \)
41 \( 1 - 1.49e5iT - 7.98e12T^{2} \)
43 \( 1 - 1.65e6T + 1.16e13T^{2} \)
47 \( 1 + 3.35e6iT - 2.38e13T^{2} \)
53 \( 1 - 2.01e6iT - 6.22e13T^{2} \)
59 \( 1 - 2.16e6iT - 1.46e14T^{2} \)
61 \( 1 + 1.66e7T + 1.91e14T^{2} \)
67 \( 1 + 2.51e7T + 4.06e14T^{2} \)
71 \( 1 - 4.99e7iT - 6.45e14T^{2} \)
73 \( 1 + 4.82e6T + 8.06e14T^{2} \)
79 \( 1 + 2.56e7T + 1.51e15T^{2} \)
83 \( 1 + 4.83e7iT - 2.25e15T^{2} \)
89 \( 1 - 3.21e6iT - 3.93e15T^{2} \)
97 \( 1 - 5.09e7T + 7.83e15T^{2} \)
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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

−12.71204619633392063507237020860, −11.92540907589542397553677139463, −10.33785054436412205265802235760, −9.756096344806845908198172379805, −8.923860914293555345555569972091, −7.36338273803259931596047794086, −4.86325183559149456353104620128, −3.58771517949170652848444911335, −2.67786152408593906868975791471, −1.44624323598003177835797704541, 0.38429903719803325911130696861, 3.05570181712684594703639343264, 4.58325796482545656220078355326, 6.10297462457138885024759077488, 7.13361491433545669281152653278, 8.059063768928530595967913384287, 9.120679045845759844952056317248, 9.932733613823099799576726085986, 12.37746970867478209586113098268, 13.56168316140343595615565282697

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