↑

Properties

Label 2-75-5.4-c13-0-17
Degree $2$
Conductor $75$
Sign $-0.894 - 0.447i$
Analytic cond. $80.4231$
Root an. cond. $8.96789$
Motivic weight $13$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 149. i·2-s − 729i·3-s − 1.40e4·4-s + 1.08e5·6-s + 3.84e5i·7-s − 8.81e5i·8-s − 5.31e5·9-s + 6.10e6·11-s + 1.02e7i·12-s − 1.13e6i·13-s − 5.73e7·14-s + 1.61e7·16-s − 2.69e7i·17-s − 7.93e7i·18-s + 3.29e8·19-s + ⋯
L(s)  = 1  + 1.64i·2-s − 0.577i·3-s − 1.72·4-s + 0.952·6-s + 1.23i·7-s − 1.18i·8-s − 0.333·9-s + 1.03·11-s + 0.993i·12-s − 0.0651i·13-s − 2.03·14-s + 0.240·16-s − 0.271i·17-s − 0.549i·18-s + 1.60·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(75\)    =    \(3 \cdot 5^{2}\)
Sign: $-0.894 - 0.447i$
Analytic conductor: \(80.4231\)
Root analytic conductor: \(8.96789\)
Motivic weight: \(13\)
Rational: no
Arithmetic: yes
Character: $\chi_{75} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 75,\ (\ :13/2),\ -0.894 - 0.447i)\)

Particular Values

\(L(7)\) \(\approx\) \(2.120782331\)
\(L(\frac12)\) \(\approx\) \(2.120782331\)
\(L(\frac{15}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 729iT \)
5 \( 1 \)
good2 \( 1 - 149. iT - 8.19e3T^{2} \)
7 \( 1 - 3.84e5iT - 9.68e10T^{2} \)
11 \( 1 - 6.10e6T + 3.45e13T^{2} \)
13 \( 1 + 1.13e6iT - 3.02e14T^{2} \)
17 \( 1 + 2.69e7iT - 9.90e15T^{2} \)
19 \( 1 - 3.29e8T + 4.20e16T^{2} \)
23 \( 1 + 4.70e8iT - 5.04e17T^{2} \)
29 \( 1 - 4.24e9T + 1.02e19T^{2} \)
31 \( 1 - 7.67e9T + 2.44e19T^{2} \)
37 \( 1 + 2.83e10iT - 2.43e20T^{2} \)
41 \( 1 + 2.11e10T + 9.25e20T^{2} \)
43 \( 1 - 6.93e10iT - 1.71e21T^{2} \)
47 \( 1 - 1.01e11iT - 5.46e21T^{2} \)
53 \( 1 - 1.81e11iT - 2.60e22T^{2} \)
59 \( 1 + 3.10e11T + 1.04e23T^{2} \)
61 \( 1 - 1.76e11T + 1.61e23T^{2} \)
67 \( 1 - 6.25e10iT - 5.48e23T^{2} \)
71 \( 1 - 1.13e12T + 1.16e24T^{2} \)
73 \( 1 - 1.17e11iT - 1.67e24T^{2} \)
79 \( 1 + 3.99e12T + 4.66e24T^{2} \)
83 \( 1 + 8.58e11iT - 8.87e24T^{2} \)
89 \( 1 - 9.67e11T + 2.19e25T^{2} \)
97 \( 1 - 2.50e12iT - 6.73e25T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.47270496241287226141986250411, −11.59879042617094545808288528829, −9.497347381881808421905042145181, −8.677188161707796059667239796387, −7.68425988204786963567656359440, −6.54970489048092819212639070805, −5.82466493585038135965767348805, −4.67081400667340329317541643082, −2.75354946860234905761544140373, −0.993421438538933513090786696459, 0.65004998814356672619263685104, 1.45180379714837819748254079063, 3.12004888419063965202318476419, 3.87999441895081321917214534400, 4.88350077426970493848136519226, 6.83704471929601183386814029911, 8.519723402772764070325474477137, 9.841735747086637722665105866858, 10.21029683171893204395532770774, 11.47941009392151703693650055517

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