Properties

Label 2-39e2-1.1-c3-0-87
Degree $2$
Conductor $1521$
Sign $1$
Analytic cond. $89.7419$
Root an. cond. $9.47322$
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  + 5.48·2-s + 22.1·4-s − 14.0·5-s − 24.2·7-s + 77.5·8-s − 77.3·10-s − 3.10·11-s − 133.·14-s + 248.·16-s + 43.9·17-s + 85.8·19-s − 311.·20-s − 17.0·22-s + 203.·23-s + 73.4·25-s − 537.·28-s − 31.0·29-s + 135.·31-s + 744.·32-s + 241.·34-s + 341.·35-s − 290.·37-s + 471.·38-s − 1.09e3·40-s + 148.·41-s + 281.·43-s − 68.7·44-s + ⋯
L(s)  = 1  + 1.94·2-s + 2.76·4-s − 1.25·5-s − 1.31·7-s + 3.42·8-s − 2.44·10-s − 0.0851·11-s − 2.54·14-s + 3.88·16-s + 0.626·17-s + 1.03·19-s − 3.48·20-s − 0.165·22-s + 1.84·23-s + 0.587·25-s − 3.62·28-s − 0.198·29-s + 0.786·31-s + 4.11·32-s + 1.21·34-s + 1.65·35-s − 1.28·37-s + 2.01·38-s − 4.31·40-s + 0.566·41-s + 0.996·43-s − 0.235·44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1521\)    =    \(3^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(89.7419\)
Root analytic conductor: \(9.47322\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1521,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(6.166696337\)
\(L(\frac12)\) \(\approx\) \(6.166696337\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
13 \( 1 \)
good2 \( 1 - 5.48T + 8T^{2} \)
5 \( 1 + 14.0T + 125T^{2} \)
7 \( 1 + 24.2T + 343T^{2} \)
11 \( 1 + 3.10T + 1.33e3T^{2} \)
17 \( 1 - 43.9T + 4.91e3T^{2} \)
19 \( 1 - 85.8T + 6.85e3T^{2} \)
23 \( 1 - 203.T + 1.21e4T^{2} \)
29 \( 1 + 31.0T + 2.43e4T^{2} \)
31 \( 1 - 135.T + 2.97e4T^{2} \)
37 \( 1 + 290.T + 5.06e4T^{2} \)
41 \( 1 - 148.T + 6.89e4T^{2} \)
43 \( 1 - 281.T + 7.95e4T^{2} \)
47 \( 1 - 225.T + 1.03e5T^{2} \)
53 \( 1 - 172.T + 1.48e5T^{2} \)
59 \( 1 - 41.2T + 2.05e5T^{2} \)
61 \( 1 - 499.T + 2.26e5T^{2} \)
67 \( 1 + 503.T + 3.00e5T^{2} \)
71 \( 1 + 946.T + 3.57e5T^{2} \)
73 \( 1 - 1.11e3T + 3.89e5T^{2} \)
79 \( 1 - 674.T + 4.93e5T^{2} \)
83 \( 1 + 59.4T + 5.71e5T^{2} \)
89 \( 1 - 1.21e3T + 7.04e5T^{2} \)
97 \( 1 - 879.T + 9.12e5T^{2} \)
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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

−9.118194601967100675315246357890, −7.77117037970789003306088778619, −7.21384510562423203304001960807, −6.57093014098471485847118510398, −5.61496657885029602100652748297, −4.87454759940361336813191133018, −3.85756569764793113997889899285, −3.34331915764237245444414103240, −2.67741733400289079674842183723, −0.911963064890950352177229371527, 0.911963064890950352177229371527, 2.67741733400289079674842183723, 3.34331915764237245444414103240, 3.85756569764793113997889899285, 4.87454759940361336813191133018, 5.61496657885029602100652748297, 6.57093014098471485847118510398, 7.21384510562423203304001960807, 7.77117037970789003306088778619, 9.118194601967100675315246357890

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