Properties

Label 2-75-3.2-c8-0-12
Degree $2$
Conductor $75$
Sign $-0.959 + 0.281i$
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  + 16.2i·2-s + (−77.7 + 22.8i)3-s − 8.51·4-s + (−371. − 1.26e3i)6-s + 569.·7-s + 4.02e3i·8-s + (5.51e3 − 3.54e3i)9-s + 6.58e3i·11-s + (661. − 194. i)12-s + 2.82e4·13-s + 9.26e3i·14-s − 6.76e4·16-s + 1.51e5i·17-s + (5.77e4 + 8.97e4i)18-s + 4.15e4·19-s + ⋯
L(s)  = 1  + 1.01i·2-s + (−0.959 + 0.281i)3-s − 0.0332·4-s + (−0.286 − 0.975i)6-s + 0.237·7-s + 0.982i·8-s + (0.841 − 0.540i)9-s + 0.449i·11-s + (0.0319 − 0.00937i)12-s + 0.987·13-s + 0.241i·14-s − 1.03·16-s + 1.80i·17-s + (0.549 + 0.855i)18-s + 0.319·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.959 + 0.281i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.959 + 0.281i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(75\)    =    \(3 \cdot 5^{2}\)
Sign: $-0.959 + 0.281i$
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: \((2,\ 75,\ (\ :4),\ -0.959 + 0.281i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.181868 - 1.26449i\)
\(L(\frac12)\) \(\approx\) \(0.181868 - 1.26449i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (77.7 - 22.8i)T \)
5 \( 1 \)
good2 \( 1 - 16.2iT - 256T^{2} \)
7 \( 1 - 569.T + 5.76e6T^{2} \)
11 \( 1 - 6.58e3iT - 2.14e8T^{2} \)
13 \( 1 - 2.82e4T + 8.15e8T^{2} \)
17 \( 1 - 1.51e5iT - 6.97e9T^{2} \)
19 \( 1 - 4.15e4T + 1.69e10T^{2} \)
23 \( 1 - 1.73e5iT - 7.83e10T^{2} \)
29 \( 1 + 7.83e5iT - 5.00e11T^{2} \)
31 \( 1 + 2.15e5T + 8.52e11T^{2} \)
37 \( 1 + 2.71e6T + 3.51e12T^{2} \)
41 \( 1 + 6.76e5iT - 7.98e12T^{2} \)
43 \( 1 - 4.21e6T + 1.16e13T^{2} \)
47 \( 1 + 8.87e6iT - 2.38e13T^{2} \)
53 \( 1 - 6.53e6iT - 6.22e13T^{2} \)
59 \( 1 - 1.54e7iT - 1.46e14T^{2} \)
61 \( 1 + 8.68e6T + 1.91e14T^{2} \)
67 \( 1 + 2.87e7T + 4.06e14T^{2} \)
71 \( 1 - 3.73e7iT - 6.45e14T^{2} \)
73 \( 1 + 3.75e7T + 8.06e14T^{2} \)
79 \( 1 - 4.22e7T + 1.51e15T^{2} \)
83 \( 1 - 7.28e7iT - 2.25e15T^{2} \)
89 \( 1 - 5.99e7iT - 3.93e15T^{2} \)
97 \( 1 - 1.15e7T + 7.83e15T^{2} \)
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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

−13.60541963534216474132898285288, −12.27083039407793569494827368671, −11.21996101353373321939972241346, −10.30883506015621430527403354437, −8.687584365096294156160595568595, −7.43474402644993770822661579863, −6.26691500199819293640340121825, −5.49603937743414621138204650226, −4.02500188546850156437383345990, −1.58763424724869127216874403707, 0.46523941589574123321205434440, 1.54969399007558032006293292932, 3.16653320088241261532215280802, 4.80836687438759945431805688629, 6.26433325658989462211366491774, 7.40254606084332952162241801562, 9.199274917543119056954515909319, 10.54851965045157321879733085543, 11.22132424056109384246882934472, 12.00888463313603645828785686031

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