Properties

Label 2-2475-1.1-c3-0-107
Degree $2$
Conductor $2475$
Sign $1$
Analytic cond. $146.029$
Root an. cond. $12.0842$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.312·2-s − 7.90·4-s + 15.1·7-s + 4.96·8-s + 11·11-s + 80.5·13-s − 4.72·14-s + 61.6·16-s + 84.3·17-s − 65.0·19-s − 3.43·22-s + 150.·23-s − 25.1·26-s − 119.·28-s + 67.6·29-s + 149.·31-s − 58.9·32-s − 26.3·34-s + 328.·37-s + 20.3·38-s − 326.·41-s + 108.·43-s − 86.9·44-s − 47.0·46-s + 573.·47-s − 114.·49-s − 636.·52-s + ⋯
L(s)  = 1  − 0.110·2-s − 0.987·4-s + 0.816·7-s + 0.219·8-s + 0.301·11-s + 1.71·13-s − 0.0901·14-s + 0.963·16-s + 1.20·17-s − 0.785·19-s − 0.0332·22-s + 1.36·23-s − 0.189·26-s − 0.806·28-s + 0.433·29-s + 0.865·31-s − 0.325·32-s − 0.132·34-s + 1.46·37-s + 0.0867·38-s − 1.24·41-s + 0.385·43-s − 0.297·44-s − 0.150·46-s + 1.78·47-s − 0.332·49-s − 1.69·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2475\)    =    \(3^{2} \cdot 5^{2} \cdot 11\)
Sign: $1$
Analytic conductor: \(146.029\)
Root analytic conductor: \(12.0842\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2475,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(2.520772455\)
\(L(\frac12)\) \(\approx\) \(2.520772455\)
\(L(\frac{5}{2})\) 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 \( 1 \)
11 \( 1 - 11T \)
good2 \( 1 + 0.312T + 8T^{2} \)
7 \( 1 - 15.1T + 343T^{2} \)
13 \( 1 - 80.5T + 2.19e3T^{2} \)
17 \( 1 - 84.3T + 4.91e3T^{2} \)
19 \( 1 + 65.0T + 6.85e3T^{2} \)
23 \( 1 - 150.T + 1.21e4T^{2} \)
29 \( 1 - 67.6T + 2.43e4T^{2} \)
31 \( 1 - 149.T + 2.97e4T^{2} \)
37 \( 1 - 328.T + 5.06e4T^{2} \)
41 \( 1 + 326.T + 6.89e4T^{2} \)
43 \( 1 - 108.T + 7.95e4T^{2} \)
47 \( 1 - 573.T + 1.03e5T^{2} \)
53 \( 1 - 412.T + 1.48e5T^{2} \)
59 \( 1 - 143.T + 2.05e5T^{2} \)
61 \( 1 - 243.T + 2.26e5T^{2} \)
67 \( 1 + 423.T + 3.00e5T^{2} \)
71 \( 1 + 396.T + 3.57e5T^{2} \)
73 \( 1 + 363.T + 3.89e5T^{2} \)
79 \( 1 + 228.T + 4.93e5T^{2} \)
83 \( 1 + 825.T + 5.71e5T^{2} \)
89 \( 1 + 1.06e3T + 7.04e5T^{2} \)
97 \( 1 - 576.T + 9.12e5T^{2} \)
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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

−8.542848730426382728412759991426, −8.123393002302787625668059444131, −7.16166000574838093858207926596, −6.09345151972350355972738953761, −5.44989797818591339241831209737, −4.51491117240492040439792472989, −3.91309011858215655738946618090, −2.92813863643258476788797143979, −1.36855037195973779902118740746, −0.854242075099329341223607798289, 0.854242075099329341223607798289, 1.36855037195973779902118740746, 2.92813863643258476788797143979, 3.91309011858215655738946618090, 4.51491117240492040439792472989, 5.44989797818591339241831209737, 6.09345151972350355972738953761, 7.16166000574838093858207926596, 8.123393002302787625668059444131, 8.542848730426382728412759991426

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