↑

Properties

Label 2-75-15.14-c8-0-17
Degree $2$
Conductor $75$
Sign $0.998 - 0.0526i$
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  + 8.27·2-s + (−39.9 − 70.4i)3-s − 187.·4-s + (−330. − 582. i)6-s + 860. i·7-s − 3.66e3·8-s + (−3.36e3 + 5.63e3i)9-s − 8.66e3i·11-s + (7.50e3 + 1.32e4i)12-s + 1.84e4i·13-s + 7.11e3i·14-s + 1.76e4·16-s − 1.10e5·17-s + (−2.78e4 + 4.65e4i)18-s + 7.12e4·19-s + ⋯
L(s)  = 1  + 0.516·2-s + (−0.493 − 0.869i)3-s − 0.732·4-s + (−0.255 − 0.449i)6-s + 0.358i·7-s − 0.895·8-s + (−0.512 + 0.858i)9-s − 0.591i·11-s + (0.361 + 0.637i)12-s + 0.646i·13-s + 0.185i·14-s + 0.269·16-s − 1.31·17-s + (−0.264 + 0.443i)18-s + 0.546·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0526i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.998 - 0.0526i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(75\)    =    \(3 \cdot 5^{2}\)
Sign: $0.998 - 0.0526i$
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: \((2,\ 75,\ (\ :4),\ 0.998 - 0.0526i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(1.32337 + 0.0348540i\)
\(L(\frac12)\) \(\approx\) \(1.32337 + 0.0348540i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (39.9 + 70.4i)T \)
5 \( 1 \)
good2 \( 1 - 8.27T + 256T^{2} \)
7 \( 1 - 860. iT - 5.76e6T^{2} \)
11 \( 1 + 8.66e3iT - 2.14e8T^{2} \)
13 \( 1 - 1.84e4iT - 8.15e8T^{2} \)
17 \( 1 + 1.10e5T + 6.97e9T^{2} \)
19 \( 1 - 7.12e4T + 1.69e10T^{2} \)
23 \( 1 - 3.69e5T + 7.83e10T^{2} \)
29 \( 1 + 3.06e5iT - 5.00e11T^{2} \)
31 \( 1 - 1.39e6T + 8.52e11T^{2} \)
37 \( 1 + 3.68e6iT - 3.51e12T^{2} \)
41 \( 1 - 5.05e6iT - 7.98e12T^{2} \)
43 \( 1 - 1.42e6iT - 1.16e13T^{2} \)
47 \( 1 + 7.79e5T + 2.38e13T^{2} \)
53 \( 1 + 7.55e6T + 6.22e13T^{2} \)
59 \( 1 - 2.08e7iT - 1.46e14T^{2} \)
61 \( 1 - 2.03e7T + 1.91e14T^{2} \)
67 \( 1 + 1.37e6iT - 4.06e14T^{2} \)
71 \( 1 - 4.19e6iT - 6.45e14T^{2} \)
73 \( 1 - 2.57e7iT - 8.06e14T^{2} \)
79 \( 1 + 1.07e7T + 1.51e15T^{2} \)
83 \( 1 - 1.63e7T + 2.25e15T^{2} \)
89 \( 1 - 5.98e7iT - 3.93e15T^{2} \)
97 \( 1 - 1.07e8iT - 7.83e15T^{2} \)
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

−13.07356702024888458418588015074, −11.95525021210115293266227198349, −11.06893180817281762500932626008, −9.303189493376932740339196753375, −8.306258664755364632403826907365, −6.75155918539992430753295211273, −5.66928660408276182085826809007, −4.48423592606255094230337459654, −2.67870753663884932667307094487, −0.844040255516190779500245363858, 0.55728655038787574866528872519, 3.17178305544519638660660307816, 4.45478625769769304857289812210, 5.22213852070074041941133742545, 6.67859769664217558726235995065, 8.551592780151436573679004691149, 9.602530461034088224427832455347, 10.59612153428255290702852515078, 11.82499592774457282427336865397, 12.93432367606988667258950004084

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