↑

Properties

Label 2-75-15.14-c8-0-31
Degree $2$
Conductor $75$
Sign $-0.836 + 0.547i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 18.2·2-s + (−40.7 + 69.9i)3-s + 75.9·4-s + (743. − 1.27e3i)6-s − 4.67e3i·7-s + 3.28e3·8-s + (−3.23e3 − 5.70e3i)9-s + 2.53e4i·11-s + (−3.09e3 + 5.31e3i)12-s − 600. i·13-s + 8.52e4i·14-s − 7.92e4·16-s + 7.32e4·17-s + (5.89e4 + 1.04e5i)18-s + 3.84e4·19-s + ⋯
L(s)  = 1  − 1.13·2-s + (−0.503 + 0.863i)3-s + 0.296·4-s + (0.573 − 0.983i)6-s − 1.94i·7-s + 0.801·8-s + (−0.492 − 0.870i)9-s + 1.72i·11-s + (−0.149 + 0.256i)12-s − 0.0210i·13-s + 2.21i·14-s − 1.20·16-s + 0.876·17-s + (0.561 + 0.990i)18-s + 0.294·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.836 + 0.547i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.836 + 0.547i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(75\)    =    \(3 \cdot 5^{2}\)
Sign: $-0.836 + 0.547i$
Analytic conductor: \(30.5533\)
Root analytic conductor: \(5.52751\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{75} (74, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 75,\ (\ :4),\ -0.836 + 0.547i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.0311216 - 0.104399i\)
\(L(\frac12)\) \(\approx\) \(0.0311216 - 0.104399i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (40.7 - 69.9i)T \)
5 \( 1 \)
good2 \( 1 + 18.2T + 256T^{2} \)
7 \( 1 + 4.67e3iT - 5.76e6T^{2} \)
11 \( 1 - 2.53e4iT - 2.14e8T^{2} \)
13 \( 1 + 600. iT - 8.15e8T^{2} \)
17 \( 1 - 7.32e4T + 6.97e9T^{2} \)
19 \( 1 - 3.84e4T + 1.69e10T^{2} \)
23 \( 1 + 1.96e5T + 7.83e10T^{2} \)
29 \( 1 - 5.04e5iT - 5.00e11T^{2} \)
31 \( 1 - 6.65e5T + 8.52e11T^{2} \)
37 \( 1 + 3.51e5iT - 3.51e12T^{2} \)
41 \( 1 + 2.88e6iT - 7.98e12T^{2} \)
43 \( 1 + 1.17e6iT - 1.16e13T^{2} \)
47 \( 1 + 4.20e6T + 2.38e13T^{2} \)
53 \( 1 - 6.57e6T + 6.22e13T^{2} \)
59 \( 1 - 1.46e7iT - 1.46e14T^{2} \)
61 \( 1 + 1.09e7T + 1.91e14T^{2} \)
67 \( 1 + 1.82e7iT - 4.06e14T^{2} \)
71 \( 1 + 1.95e7iT - 6.45e14T^{2} \)
73 \( 1 + 3.13e7iT - 8.06e14T^{2} \)
79 \( 1 - 1.63e6T + 1.51e15T^{2} \)
83 \( 1 + 3.92e7T + 2.25e15T^{2} \)
89 \( 1 + 1.89e7iT - 3.93e15T^{2} \)
97 \( 1 - 1.16e8iT - 7.83e15T^{2} \)
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

−12.13438058174775072905425749487, −10.63287876010722044032399740632, −10.19443891957092636691578672969, −9.428546019925272389597377912401, −7.78736026773105375671958458781, −6.93152810935761364515794594881, −4.81835882374000337285859730833, −3.91220136530220446694497165307, −1.28878901735383688260322029471, −0.06542864819627830699689198340, 1.25557363043736149569666549996, 2.69666961979635877137413295351, 5.38784382050710803725173129486, 6.26555253631371288624722769216, 8.062471987401214970034768308717, 8.508036057625852987912260128353, 9.759616881276896050551171086593, 11.25777519317098983603431906559, 11.92156176556690349658666760107, 13.17135551401498652271001165188

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