Properties

Label 2-75-1.1-c21-0-13
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  − 1.72e3·2-s + 5.90e4·3-s + 8.79e5·4-s − 1.01e8·6-s + 4.99e8·7-s + 2.10e9·8-s + 3.48e9·9-s − 1.14e11·11-s + 5.19e10·12-s + 1.56e11·13-s − 8.62e11·14-s − 5.46e12·16-s − 2.48e12·17-s − 6.01e12·18-s − 4.52e13·19-s + 2.95e13·21-s + 1.97e14·22-s + 6.05e12·23-s + 1.24e14·24-s − 2.69e14·26-s + 2.05e14·27-s + 4.39e14·28-s − 2.25e15·29-s + 2.57e15·31-s + 5.03e15·32-s − 6.75e15·33-s + 4.29e15·34-s + ⋯
L(s)  = 1  − 1.19·2-s + 0.577·3-s + 0.419·4-s − 0.687·6-s + 0.669·7-s + 0.691·8-s + 0.333·9-s − 1.32·11-s + 0.242·12-s + 0.314·13-s − 0.797·14-s − 1.24·16-s − 0.299·17-s − 0.397·18-s − 1.69·19-s + 0.386·21-s + 1.58·22-s + 0.0304·23-s + 0.399·24-s − 0.374·26-s + 0.192·27-s + 0.280·28-s − 0.993·29-s + 0.564·31-s + 0.789·32-s − 0.767·33-s + 0.356·34-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

\(L(11)\) \(\approx\) \(0.9360596404\)
\(L(\frac12)\) \(\approx\) \(0.9360596404\)
\(L(\frac{23}{2})\) 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 \( 1 + 1.72e3T + 2.09e6T^{2} \)
7 \( 1 - 4.99e8T + 5.58e17T^{2} \)
11 \( 1 + 1.14e11T + 7.40e21T^{2} \)
13 \( 1 - 1.56e11T + 2.47e23T^{2} \)
17 \( 1 + 2.48e12T + 6.90e25T^{2} \)
19 \( 1 + 4.52e13T + 7.14e26T^{2} \)
23 \( 1 - 6.05e12T + 3.94e28T^{2} \)
29 \( 1 + 2.25e15T + 5.13e30T^{2} \)
31 \( 1 - 2.57e15T + 2.08e31T^{2} \)
37 \( 1 - 2.36e16T + 8.55e32T^{2} \)
41 \( 1 + 7.10e16T + 7.38e33T^{2} \)
43 \( 1 + 1.38e17T + 2.00e34T^{2} \)
47 \( 1 - 5.07e17T + 1.30e35T^{2} \)
53 \( 1 + 1.50e18T + 1.62e36T^{2} \)
59 \( 1 - 2.27e18T + 1.54e37T^{2} \)
61 \( 1 - 5.10e18T + 3.10e37T^{2} \)
67 \( 1 - 2.36e19T + 2.22e38T^{2} \)
71 \( 1 - 1.27e19T + 7.52e38T^{2} \)
73 \( 1 - 2.08e19T + 1.34e39T^{2} \)
79 \( 1 - 3.58e19T + 7.08e39T^{2} \)
83 \( 1 - 1.35e20T + 1.99e40T^{2} \)
89 \( 1 + 1.87e20T + 8.65e40T^{2} \)
97 \( 1 + 6.13e20T + 5.27e41T^{2} \)
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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

−10.47891556874165598648143077851, −9.451081820139773313926918361757, −8.302951636816039982282047068164, −8.038089236008669560526843426105, −6.76097035718912021811282166038, −5.12340260118711975410774339306, −4.06506160768580158953291247013, −2.46591410477533348818754496255, −1.71505866093063698046508024531, −0.46486651553146876163237227958, 0.46486651553146876163237227958, 1.71505866093063698046508024531, 2.46591410477533348818754496255, 4.06506160768580158953291247013, 5.12340260118711975410774339306, 6.76097035718912021811282166038, 8.038089236008669560526843426105, 8.302951636816039982282047068164, 9.451081820139773313926918361757, 10.47891556874165598648143077851

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