Properties

Label 2-75-1.1-c21-0-21
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 625.·2-s + 5.90e4·3-s − 1.70e6·4-s − 3.69e7·6-s − 3.78e8·7-s + 2.37e9·8-s + 3.48e9·9-s − 3.73e10·11-s − 1.00e11·12-s + 9.27e11·13-s + 2.36e11·14-s + 2.09e12·16-s + 1.40e13·17-s − 2.18e12·18-s + 3.31e13·19-s − 2.23e13·21-s + 2.33e13·22-s − 3.30e14·23-s + 1.40e14·24-s − 5.79e14·26-s + 2.05e14·27-s + 6.46e14·28-s + 2.35e15·29-s + 3.97e15·31-s − 6.29e15·32-s − 2.20e15·33-s − 8.77e15·34-s + ⋯
L(s)  = 1  − 0.431·2-s + 0.577·3-s − 0.813·4-s − 0.249·6-s − 0.506·7-s + 0.783·8-s + 0.333·9-s − 0.433·11-s − 0.469·12-s + 1.86·13-s + 0.218·14-s + 0.475·16-s + 1.68·17-s − 0.143·18-s + 1.23·19-s − 0.292·21-s + 0.187·22-s − 1.66·23-s + 0.452·24-s − 0.805·26-s + 0.192·27-s + 0.412·28-s + 1.04·29-s + 0.870·31-s − 0.988·32-s − 0.250·33-s − 0.729·34-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: \(0\)
Selberg data: \((2,\ 75,\ (\ :21/2),\ 1)\)

Particular Values

\(L(11)\) \(\approx\) \(2.132002004\)
\(L(\frac12)\) \(\approx\) \(2.132002004\)
\(L(\frac{23}{2})\) 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 \( 1 + 625.T + 2.09e6T^{2} \)
7 \( 1 + 3.78e8T + 5.58e17T^{2} \)
11 \( 1 + 3.73e10T + 7.40e21T^{2} \)
13 \( 1 - 9.27e11T + 2.47e23T^{2} \)
17 \( 1 - 1.40e13T + 6.90e25T^{2} \)
19 \( 1 - 3.31e13T + 7.14e26T^{2} \)
23 \( 1 + 3.30e14T + 3.94e28T^{2} \)
29 \( 1 - 2.35e15T + 5.13e30T^{2} \)
31 \( 1 - 3.97e15T + 2.08e31T^{2} \)
37 \( 1 - 4.11e16T + 8.55e32T^{2} \)
41 \( 1 + 5.23e16T + 7.38e33T^{2} \)
43 \( 1 + 2.10e17T + 2.00e34T^{2} \)
47 \( 1 + 5.77e17T + 1.30e35T^{2} \)
53 \( 1 + 1.16e18T + 1.62e36T^{2} \)
59 \( 1 - 4.17e18T + 1.54e37T^{2} \)
61 \( 1 - 9.77e18T + 3.10e37T^{2} \)
67 \( 1 - 1.53e19T + 2.22e38T^{2} \)
71 \( 1 + 6.75e18T + 7.52e38T^{2} \)
73 \( 1 - 1.58e19T + 1.34e39T^{2} \)
79 \( 1 + 1.48e19T + 7.08e39T^{2} \)
83 \( 1 - 6.30e19T + 1.99e40T^{2} \)
89 \( 1 - 1.22e20T + 8.65e40T^{2} \)
97 \( 1 + 5.41e20T + 5.27e41T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.06554428549136163462493952707, −9.754050224771117892076275995468, −8.282337802011442035470211316656, −8.018814247650825381964538358578, −6.35386830467607815544680369770, −5.19237455553997743869634004930, −3.81276074114955752671402312844, −3.15877610266214529398002195473, −1.46592929815049956714237597212, −0.70407349876262103304756547841, 0.70407349876262103304756547841, 1.46592929815049956714237597212, 3.15877610266214529398002195473, 3.81276074114955752671402312844, 5.19237455553997743869634004930, 6.35386830467607815544680369770, 8.018814247650825381964538358578, 8.282337802011442035470211316656, 9.754050224771117892076275995468, 10.06554428549136163462493952707

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