Properties

Label 2-75-3.2-c8-0-18
Degree $2$
Conductor $75$
Sign $-0.180 - 0.983i$
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  − 3.03i·2-s + (14.6 + 79.6i)3-s + 246.·4-s + (241. − 44.3i)6-s + 59.7·7-s − 1.52e3i·8-s + (−6.13e3 + 2.32e3i)9-s + 2.18e4i·11-s + (3.60e3 + 1.96e4i)12-s + 3.84e4·13-s − 181. i·14-s + 5.85e4·16-s − 3.61e4i·17-s + (7.06e3 + 1.86e4i)18-s − 1.36e5·19-s + ⋯
L(s)  = 1  − 0.189i·2-s + (0.180 + 0.983i)3-s + 0.964·4-s + (0.186 − 0.0341i)6-s + 0.0248·7-s − 0.372i·8-s + (−0.934 + 0.354i)9-s + 1.49i·11-s + (0.173 + 0.948i)12-s + 1.34·13-s − 0.00471i·14-s + 0.893·16-s − 0.432i·17-s + (0.0672 + 0.177i)18-s − 1.04·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.180 - 0.983i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.180 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(75\)    =    \(3 \cdot 5^{2}\)
Sign: $-0.180 - 0.983i$
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: \((2,\ 75,\ (\ :4),\ -0.180 - 0.983i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(1.58523 + 1.90219i\)
\(L(\frac12)\) \(\approx\) \(1.58523 + 1.90219i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-14.6 - 79.6i)T \)
5 \( 1 \)
good2 \( 1 + 3.03iT - 256T^{2} \)
7 \( 1 - 59.7T + 5.76e6T^{2} \)
11 \( 1 - 2.18e4iT - 2.14e8T^{2} \)
13 \( 1 - 3.84e4T + 8.15e8T^{2} \)
17 \( 1 + 3.61e4iT - 6.97e9T^{2} \)
19 \( 1 + 1.36e5T + 1.69e10T^{2} \)
23 \( 1 - 2.35e5iT - 7.83e10T^{2} \)
29 \( 1 - 8.96e5iT - 5.00e11T^{2} \)
31 \( 1 - 6.80e4T + 8.52e11T^{2} \)
37 \( 1 - 6.65e4T + 3.51e12T^{2} \)
41 \( 1 - 3.76e6iT - 7.98e12T^{2} \)
43 \( 1 - 2.03e6T + 1.16e13T^{2} \)
47 \( 1 + 5.93e6iT - 2.38e13T^{2} \)
53 \( 1 - 1.31e7iT - 6.22e13T^{2} \)
59 \( 1 - 1.58e7iT - 1.46e14T^{2} \)
61 \( 1 + 3.51e6T + 1.91e14T^{2} \)
67 \( 1 + 1.39e7T + 4.06e14T^{2} \)
71 \( 1 - 1.51e7iT - 6.45e14T^{2} \)
73 \( 1 + 3.93e7T + 8.06e14T^{2} \)
79 \( 1 - 3.15e7T + 1.51e15T^{2} \)
83 \( 1 + 4.88e7iT - 2.25e15T^{2} \)
89 \( 1 + 1.09e8iT - 3.93e15T^{2} \)
97 \( 1 - 1.16e8T + 7.83e15T^{2} \)
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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

−13.09691502957546885215222748672, −11.83138246278610226154956896367, −10.86765646537068428569779150499, −10.04298417484486994986508282647, −8.814415746881881218608562941192, −7.36759068401852681200527282549, −6.00939386236860191747780210836, −4.46408015536881653776175624765, −3.13644420483490294925533880737, −1.69748173813427837275796391554, 0.72698544528957447495255407896, 2.09846209206546511926515277591, 3.44544693627890888442565112772, 5.98417276423422344432600381870, 6.43977204899651374262564895570, 7.977131226542779825741435669721, 8.649528218902472273887054470972, 10.77325309268673225223003652292, 11.40230335471590493946377666047, 12.62752317052843445144687424900

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