↑

Properties

Label 2-75-5.4-c21-0-58
Degree $2$
Conductor $75$
Sign $-0.894 + 0.447i$
Analytic cond. $209.608$
Root an. cond. $14.4778$
Motivic weight $21$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 544i·2-s + 5.90e4i·3-s + 1.80e6·4-s + 3.21e7·6-s − 1.27e9i·7-s − 2.12e9i·8-s − 3.48e9·9-s − 7.75e10·11-s + 1.06e11i·12-s − 4.34e11i·13-s − 6.95e11·14-s + 2.62e12·16-s − 1.28e13i·17-s + 1.89e12i·18-s + 2.86e13·19-s + ⋯
L(s)  = 1  − 0.375i·2-s + 0.577i·3-s + 0.858·4-s + 0.216·6-s − 1.70i·7-s − 0.698i·8-s − 0.333·9-s − 0.901·11-s + 0.495i·12-s − 0.873i·13-s − 0.642·14-s + 0.596·16-s − 1.54i·17-s + 0.125i·18-s + 1.07·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(75\)    =    \(3 \cdot 5^{2}\)
Sign: $-0.894 + 0.447i$
Analytic conductor: \(209.608\)
Root analytic conductor: \(14.4778\)
Motivic weight: \(21\)
Rational: no
Arithmetic: yes
Character: $\chi_{75} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 75,\ (\ :21/2),\ -0.894 + 0.447i)\)

Particular Values

\(L(11)\) \(\approx\) \(2.419965240\)
\(L(\frac12)\) \(\approx\) \(2.419965240\)
\(L(\frac{23}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 5.90e4iT \)
5 \( 1 \)
good2 \( 1 + 544iT - 2.09e6T^{2} \)
7 \( 1 + 1.27e9iT - 5.58e17T^{2} \)
11 \( 1 + 7.75e10T + 7.40e21T^{2} \)
13 \( 1 + 4.34e11iT - 2.47e23T^{2} \)
17 \( 1 + 1.28e13iT - 6.90e25T^{2} \)
19 \( 1 - 2.86e13T + 7.14e26T^{2} \)
23 \( 1 - 2.24e14iT - 3.94e28T^{2} \)
29 \( 1 - 5.16e13T + 5.13e30T^{2} \)
31 \( 1 - 8.92e15T + 2.08e31T^{2} \)
37 \( 1 + 4.39e16iT - 8.55e32T^{2} \)
41 \( 1 - 5.81e16T + 7.38e33T^{2} \)
43 \( 1 + 1.61e17iT - 2.00e34T^{2} \)
47 \( 1 - 1.60e17iT - 1.30e35T^{2} \)
53 \( 1 - 2.29e18iT - 1.62e36T^{2} \)
59 \( 1 + 5.15e18T + 1.54e37T^{2} \)
61 \( 1 - 1.25e18T + 3.10e37T^{2} \)
67 \( 1 - 5.40e18iT - 2.22e38T^{2} \)
71 \( 1 + 1.10e19T + 7.52e38T^{2} \)
73 \( 1 + 3.77e19iT - 1.34e39T^{2} \)
79 \( 1 + 6.31e19T + 7.08e39T^{2} \)
83 \( 1 + 1.45e20iT - 1.99e40T^{2} \)
89 \( 1 + 1.37e20T + 8.65e40T^{2} \)
97 \( 1 - 3.24e20iT - 5.27e41T^{2} \)
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

−10.36042220760514300015902062344, −9.582957060177314940491300022074, −7.63616960798431706065656080745, −7.31842652121896445789252854343, −5.75224585574992692262112503065, −4.59071811986835303198888610619, −3.38333037202406607495552751445, −2.71105110265892208653284275635, −1.10151404710942036482373667321, −0.42319798172835665039815170535, 1.36865520901347859720162580677, 2.30717519189310408124077141484, 2.95103798589061202117787510295, 4.93842359767822846108341535453, 6.01059813443408068690578454424, 6.56264906461255927839662847157, 7.998880043031489796676519571838, 8.541235757411729844553343037071, 10.03852225109877040080750869067, 11.38234636389406829678087864220

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