Properties

Label 2-75-1.1-c21-0-54
Degree $2$
Conductor $75$
Sign $-1$
Analytic cond. $209.608$
Root an. cond. $14.4778$
Motivic weight $21$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 1.57e3·2-s − 5.90e4·3-s + 3.70e5·4-s − 9.27e7·6-s + 1.39e9·7-s − 2.71e9·8-s + 3.48e9·9-s − 1.07e11·11-s − 2.18e10·12-s + 6.34e11·13-s + 2.19e12·14-s − 5.03e12·16-s − 9.95e12·17-s + 5.47e12·18-s + 2.82e12·19-s − 8.24e13·21-s − 1.69e14·22-s + 7.33e13·23-s + 1.60e14·24-s + 9.95e14·26-s − 2.05e14·27-s + 5.17e14·28-s − 1.24e15·29-s + 6.63e15·31-s − 2.22e15·32-s + 6.36e15·33-s − 1.56e16·34-s + ⋯
L(s)  = 1  + 1.08·2-s − 0.577·3-s + 0.176·4-s − 0.626·6-s + 1.86·7-s − 0.893·8-s + 0.333·9-s − 1.25·11-s − 0.101·12-s + 1.27·13-s + 2.02·14-s − 1.14·16-s − 1.19·17-s + 0.361·18-s + 0.105·19-s − 1.07·21-s − 1.35·22-s + 0.369·23-s + 0.515·24-s + 1.38·26-s − 0.192·27-s + 0.329·28-s − 0.549·29-s + 1.45·31-s − 0.349·32-s + 0.723·33-s − 1.29·34-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(75\)    =    \(3 \cdot 5^{2}\)
Sign: $-1$
Analytic conductor: \(209.608\)
Root analytic conductor: \(14.4778\)
Motivic weight: \(21\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 75,\ (\ :21/2),\ -1)\)

Particular Values

\(L(11)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{23}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 5.90e4T \)
5 \( 1 \)
good2 \( 1 - 1.57e3T + 2.09e6T^{2} \)
7 \( 1 - 1.39e9T + 5.58e17T^{2} \)
11 \( 1 + 1.07e11T + 7.40e21T^{2} \)
13 \( 1 - 6.34e11T + 2.47e23T^{2} \)
17 \( 1 + 9.95e12T + 6.90e25T^{2} \)
19 \( 1 - 2.82e12T + 7.14e26T^{2} \)
23 \( 1 - 7.33e13T + 3.94e28T^{2} \)
29 \( 1 + 1.24e15T + 5.13e30T^{2} \)
31 \( 1 - 6.63e15T + 2.08e31T^{2} \)
37 \( 1 + 5.11e16T + 8.55e32T^{2} \)
41 \( 1 - 5.89e16T + 7.38e33T^{2} \)
43 \( 1 + 1.83e17T + 2.00e34T^{2} \)
47 \( 1 - 4.57e16T + 1.30e35T^{2} \)
53 \( 1 - 1.30e18T + 1.62e36T^{2} \)
59 \( 1 + 5.67e18T + 1.54e37T^{2} \)
61 \( 1 + 7.20e18T + 3.10e37T^{2} \)
67 \( 1 - 2.10e18T + 2.22e38T^{2} \)
71 \( 1 - 4.72e19T + 7.52e38T^{2} \)
73 \( 1 - 5.09e19T + 1.34e39T^{2} \)
79 \( 1 - 7.07e18T + 7.08e39T^{2} \)
83 \( 1 + 5.62e19T + 1.99e40T^{2} \)
89 \( 1 - 1.35e20T + 8.65e40T^{2} \)
97 \( 1 - 6.40e19T + 5.27e41T^{2} \)
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

−10.66061644568953430829783123986, −8.823789528675921971372049290473, −7.966312352098962177493991475352, −6.50718408720235023312585211046, −5.32269986807292509432800869227, −4.86962128316358781525345112834, −3.89389040332164542194620265297, −2.45359339868058502071877342299, −1.30994891143682443975852466345, 0, 1.30994891143682443975852466345, 2.45359339868058502071877342299, 3.89389040332164542194620265297, 4.86962128316358781525345112834, 5.32269986807292509432800869227, 6.50718408720235023312585211046, 7.966312352098962177493991475352, 8.823789528675921971372049290473, 10.66061644568953430829783123986

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