Properties

Label 2-675-1.1-c3-0-43
Degree $2$
Conductor $675$
Sign $1$
Analytic cond. $39.8262$
Root an. cond. $6.31080$
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  + 4.90·2-s + 16.1·4-s + 2.10·7-s + 39.7·8-s − 40.3·11-s + 61.7·13-s + 10.3·14-s + 66.5·16-s + 99.2·17-s + 134.·19-s − 197.·22-s + 76.4·23-s + 303.·26-s + 33.8·28-s − 236.·29-s + 243.·31-s + 8.27·32-s + 487.·34-s + 57.4·37-s + 660.·38-s + 411.·41-s + 102.·43-s − 649.·44-s + 375.·46-s − 260.·47-s − 338.·49-s + 994.·52-s + ⋯
L(s)  = 1  + 1.73·2-s + 2.01·4-s + 0.113·7-s + 1.75·8-s − 1.10·11-s + 1.31·13-s + 0.197·14-s + 1.03·16-s + 1.41·17-s + 1.62·19-s − 1.91·22-s + 0.693·23-s + 2.28·26-s + 0.228·28-s − 1.51·29-s + 1.41·31-s + 0.0457·32-s + 2.45·34-s + 0.255·37-s + 2.81·38-s + 1.56·41-s + 0.364·43-s − 2.22·44-s + 1.20·46-s − 0.808·47-s − 0.987·49-s + 2.65·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(675\)    =    \(3^{3} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(39.8262\)
Root analytic conductor: \(6.31080\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 675,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(6.395371400\)
\(L(\frac12)\) \(\approx\) \(6.395371400\)
\(L(\frac{5}{2})\) 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 \( 1 \)
good2 \( 1 - 4.90T + 8T^{2} \)
7 \( 1 - 2.10T + 343T^{2} \)
11 \( 1 + 40.3T + 1.33e3T^{2} \)
13 \( 1 - 61.7T + 2.19e3T^{2} \)
17 \( 1 - 99.2T + 4.91e3T^{2} \)
19 \( 1 - 134.T + 6.85e3T^{2} \)
23 \( 1 - 76.4T + 1.21e4T^{2} \)
29 \( 1 + 236.T + 2.43e4T^{2} \)
31 \( 1 - 243.T + 2.97e4T^{2} \)
37 \( 1 - 57.4T + 5.06e4T^{2} \)
41 \( 1 - 411.T + 6.89e4T^{2} \)
43 \( 1 - 102.T + 7.95e4T^{2} \)
47 \( 1 + 260.T + 1.03e5T^{2} \)
53 \( 1 - 75.4T + 1.48e5T^{2} \)
59 \( 1 - 39.2T + 2.05e5T^{2} \)
61 \( 1 + 675.T + 2.26e5T^{2} \)
67 \( 1 + 601.T + 3.00e5T^{2} \)
71 \( 1 + 222.T + 3.57e5T^{2} \)
73 \( 1 + 297.T + 3.89e5T^{2} \)
79 \( 1 + 95.6T + 4.93e5T^{2} \)
83 \( 1 + 3.08T + 5.71e5T^{2} \)
89 \( 1 + 1.34e3T + 7.04e5T^{2} \)
97 \( 1 - 1.48e3T + 9.12e5T^{2} \)
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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

−10.45972375847541759333591412607, −9.361166784512038892029143954710, −7.953565604520043937665312167304, −7.33491326343232940758454087937, −6.03291985119077126897523238088, −5.54421701955424377785599086218, −4.64919964007318686249491714894, −3.46869586077622445782118008991, −2.86162708214065571049961306281, −1.28298752287555848132849144909, 1.28298752287555848132849144909, 2.86162708214065571049961306281, 3.46869586077622445782118008991, 4.64919964007318686249491714894, 5.54421701955424377785599086218, 6.03291985119077126897523238088, 7.33491326343232940758454087937, 7.953565604520043937665312167304, 9.361166784512038892029143954710, 10.45972375847541759333591412607

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