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Properties

Label 2-75-5.4-c9-0-3
Degree $2$
Conductor $75$
Sign $0.894 - 0.447i$
Analytic cond. $38.6276$
Root an. cond. $6.21511$
Motivic weight $9$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 22i·2-s − 81i·3-s + 28·4-s − 1.78e3·6-s + 5.98e3i·7-s − 1.18e4i·8-s − 6.56e3·9-s − 1.46e4·11-s − 2.26e3i·12-s + 3.79e4i·13-s + 1.31e5·14-s − 2.47e5·16-s + 4.41e5i·17-s + 1.44e5i·18-s − 4.41e5·19-s + ⋯
L(s)  = 1  − 0.972i·2-s − 0.577i·3-s + 0.0546·4-s − 0.561·6-s + 0.942i·7-s − 1.02i·8-s − 0.333·9-s − 0.301·11-s − 0.0315i·12-s + 0.368i·13-s + 0.916·14-s − 0.942·16-s + 1.28i·17-s + 0.324i·18-s − 0.777·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(75\)    =    \(3 \cdot 5^{2}\)
Sign: $0.894 - 0.447i$
Analytic conductor: \(38.6276\)
Root analytic conductor: \(6.21511\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{75} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 75,\ (\ :9/2),\ 0.894 - 0.447i)\)

Particular Values

\(L(5)\) \(\approx\) \(1.21502 + 0.286827i\)
\(L(\frac12)\) \(\approx\) \(1.21502 + 0.286827i\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 81iT \)
5 \( 1 \)
good2 \( 1 + 22iT - 512T^{2} \)
7 \( 1 - 5.98e3iT - 4.03e7T^{2} \)
11 \( 1 + 1.46e4T + 2.35e9T^{2} \)
13 \( 1 - 3.79e4iT - 1.06e10T^{2} \)
17 \( 1 - 4.41e5iT - 1.18e11T^{2} \)
19 \( 1 + 4.41e5T + 3.22e11T^{2} \)
23 \( 1 - 2.26e6iT - 1.80e12T^{2} \)
29 \( 1 - 1.04e6T + 1.45e13T^{2} \)
31 \( 1 + 7.91e6T + 2.64e13T^{2} \)
37 \( 1 - 2.09e7iT - 1.29e14T^{2} \)
41 \( 1 - 1.32e7T + 3.27e14T^{2} \)
43 \( 1 + 2.31e7iT - 5.02e14T^{2} \)
47 \( 1 - 1.38e7iT - 1.11e15T^{2} \)
53 \( 1 + 5.76e7iT - 3.29e15T^{2} \)
59 \( 1 - 3.20e7T + 8.66e15T^{2} \)
61 \( 1 - 1.10e8T + 1.16e16T^{2} \)
67 \( 1 - 1.18e8iT - 2.72e16T^{2} \)
71 \( 1 - 2.76e8T + 4.58e16T^{2} \)
73 \( 1 + 2.64e8iT - 5.88e16T^{2} \)
79 \( 1 + 4.48e8T + 1.19e17T^{2} \)
83 \( 1 - 8.51e8iT - 1.86e17T^{2} \)
89 \( 1 + 1.89e8T + 3.50e17T^{2} \)
97 \( 1 - 1.01e9iT - 7.60e17T^{2} \)
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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

−12.56684008616730797580044102699, −11.71701503042245599847971439042, −10.80053147684857431110336739471, −9.535362288367991398193201668266, −8.300180848530345090116581746499, −6.85437863371775176420743866317, −5.64962045413972067982749364013, −3.70580092723797461145831204968, −2.34731343852841413103705855525, −1.46165159683835399454857114661, 0.33832489089603146709014820786, 2.54397704664273585491582810438, 4.27320938169622283223041379902, 5.48028155969253773839515635547, 6.80336810253944511128318883854, 7.75327791669520941189934943986, 8.986513306100013007822394697892, 10.44890394265082589090948200058, 11.18118239488921377651088810580, 12.74594626813386424008245941076

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