Properties

Label 2-2475-1.1-c3-0-14
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  − 2.41·2-s − 2.16·4-s − 35.7·7-s + 24.5·8-s − 11·11-s + 41.3·13-s + 86.4·14-s − 41.9·16-s + 13.1·17-s − 30.3·19-s + 26.5·22-s − 18.0·23-s − 99.7·26-s + 77.6·28-s − 106.·29-s + 125.·31-s − 95.2·32-s − 31.8·34-s − 71.8·37-s + 73.1·38-s + 203.·41-s − 283.·43-s + 23.8·44-s + 43.5·46-s − 60.0·47-s + 938.·49-s − 89.6·52-s + ⋯
L(s)  = 1  − 0.853·2-s − 0.271·4-s − 1.93·7-s + 1.08·8-s − 0.301·11-s + 0.881·13-s + 1.65·14-s − 0.655·16-s + 0.188·17-s − 0.365·19-s + 0.257·22-s − 0.163·23-s − 0.752·26-s + 0.524·28-s − 0.683·29-s + 0.727·31-s − 0.525·32-s − 0.160·34-s − 0.319·37-s + 0.312·38-s + 0.775·41-s − 1.00·43-s + 0.0817·44-s + 0.139·46-s − 0.186·47-s + 2.73·49-s − 0.239·52-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

\(L(2)\) \(\approx\) \(0.3659664015\)
\(L(\frac12)\) \(\approx\) \(0.3659664015\)
\(L(\frac{5}{2})\) 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 \( 1 + 2.41T + 8T^{2} \)
7 \( 1 + 35.7T + 343T^{2} \)
13 \( 1 - 41.3T + 2.19e3T^{2} \)
17 \( 1 - 13.1T + 4.91e3T^{2} \)
19 \( 1 + 30.3T + 6.85e3T^{2} \)
23 \( 1 + 18.0T + 1.21e4T^{2} \)
29 \( 1 + 106.T + 2.43e4T^{2} \)
31 \( 1 - 125.T + 2.97e4T^{2} \)
37 \( 1 + 71.8T + 5.06e4T^{2} \)
41 \( 1 - 203.T + 6.89e4T^{2} \)
43 \( 1 + 283.T + 7.95e4T^{2} \)
47 \( 1 + 60.0T + 1.03e5T^{2} \)
53 \( 1 + 352.T + 1.48e5T^{2} \)
59 \( 1 + 9.68T + 2.05e5T^{2} \)
61 \( 1 + 200.T + 2.26e5T^{2} \)
67 \( 1 + 1.02e3T + 3.00e5T^{2} \)
71 \( 1 + 1.02e3T + 3.57e5T^{2} \)
73 \( 1 - 411.T + 3.89e5T^{2} \)
79 \( 1 + 790.T + 4.93e5T^{2} \)
83 \( 1 + 666.T + 5.71e5T^{2} \)
89 \( 1 + 496.T + 7.04e5T^{2} \)
97 \( 1 - 1.40e3T + 9.12e5T^{2} \)
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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

−8.819887367134401613353421773952, −7.938775656856018597322986973773, −7.14557439015728789658310988217, −6.34622949879230183249123334036, −5.70425260514419102828067672247, −4.47885043441857919289595998516, −3.63497231211987200700872471291, −2.83719727514061050367060988467, −1.46772490343500250360342531015, −0.31693162375796935246456005950, 0.31693162375796935246456005950, 1.46772490343500250360342531015, 2.83719727514061050367060988467, 3.63497231211987200700872471291, 4.47885043441857919289595998516, 5.70425260514419102828067672247, 6.34622949879230183249123334036, 7.14557439015728789658310988217, 7.938775656856018597322986973773, 8.819887367134401613353421773952

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