Properties

Label 2-75-1.1-c21-0-28
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  − 9.99e2·2-s − 5.90e4·3-s − 1.09e6·4-s + 5.90e7·6-s − 9.82e7·7-s + 3.19e9·8-s + 3.48e9·9-s − 9.60e10·11-s + 6.48e10·12-s − 6.22e11·13-s + 9.82e10·14-s − 8.91e11·16-s + 1.42e13·17-s − 3.48e12·18-s − 8.28e12·19-s + 5.80e12·21-s + 9.60e13·22-s + 1.17e14·23-s − 1.88e14·24-s + 6.21e14·26-s − 2.05e14·27-s + 1.07e14·28-s − 1.98e15·29-s − 5.31e15·31-s − 5.80e15·32-s + 5.67e15·33-s − 1.42e16·34-s + ⋯
L(s)  = 1  − 0.690·2-s − 0.577·3-s − 0.523·4-s + 0.398·6-s − 0.131·7-s + 1.05·8-s + 0.333·9-s − 1.11·11-s + 0.302·12-s − 1.25·13-s + 0.0907·14-s − 0.202·16-s + 1.71·17-s − 0.230·18-s − 0.310·19-s + 0.0758·21-s + 0.771·22-s + 0.591·23-s − 0.607·24-s + 0.864·26-s − 0.192·27-s + 0.0687·28-s − 0.877·29-s − 1.16·31-s − 0.911·32-s + 0.644·33-s − 1.18·34-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 \( 1 + 9.99e2T + 2.09e6T^{2} \)
7 \( 1 + 9.82e7T + 5.58e17T^{2} \)
11 \( 1 + 9.60e10T + 7.40e21T^{2} \)
13 \( 1 + 6.22e11T + 2.47e23T^{2} \)
17 \( 1 - 1.42e13T + 6.90e25T^{2} \)
19 \( 1 + 8.28e12T + 7.14e26T^{2} \)
23 \( 1 - 1.17e14T + 3.94e28T^{2} \)
29 \( 1 + 1.98e15T + 5.13e30T^{2} \)
31 \( 1 + 5.31e15T + 2.08e31T^{2} \)
37 \( 1 + 1.99e16T + 8.55e32T^{2} \)
41 \( 1 + 2.99e16T + 7.38e33T^{2} \)
43 \( 1 - 3.44e16T + 2.00e34T^{2} \)
47 \( 1 - 5.23e17T + 1.30e35T^{2} \)
53 \( 1 - 1.54e18T + 1.62e36T^{2} \)
59 \( 1 + 6.81e17T + 1.54e37T^{2} \)
61 \( 1 + 3.28e18T + 3.10e37T^{2} \)
67 \( 1 + 1.07e19T + 2.22e38T^{2} \)
71 \( 1 - 3.71e19T + 7.52e38T^{2} \)
73 \( 1 - 6.79e19T + 1.34e39T^{2} \)
79 \( 1 - 1.28e20T + 7.08e39T^{2} \)
83 \( 1 + 1.24e20T + 1.99e40T^{2} \)
89 \( 1 - 4.94e20T + 8.65e40T^{2} \)
97 \( 1 + 4.69e20T + 5.27e41T^{2} \)
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

−10.08386317434043130255547046666, −9.226434097597783526434955504308, −7.87206648912958637161171157265, −7.25352453154079232782140459795, −5.52267417409517508723218044831, −4.93856447043331759867500412093, −3.53726703008427095111706237110, −2.08323946446096562469763829785, −0.802520746609033305877807490256, 0, 0.802520746609033305877807490256, 2.08323946446096562469763829785, 3.53726703008427095111706237110, 4.93856447043331759867500412093, 5.52267417409517508723218044831, 7.25352453154079232782140459795, 7.87206648912958637161171157265, 9.226434097597783526434955504308, 10.08386317434043130255547046666

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