Properties

Label 2-1323-1.1-c3-0-75
Degree $2$
Conductor $1323$
Sign $1$
Analytic cond. $78.0595$
Root an. cond. $8.83513$
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  + 5.15·2-s + 18.5·4-s − 17.0·5-s + 54.6·8-s − 87.7·10-s − 30.3·11-s + 89.5·13-s + 132.·16-s − 13.4·17-s + 5.37·19-s − 316.·20-s − 156.·22-s + 167.·23-s + 164.·25-s + 461.·26-s − 135.·29-s + 18.9·31-s + 248.·32-s − 69.1·34-s + 402.·37-s + 27.7·38-s − 929.·40-s + 434.·41-s + 53.1·43-s − 564.·44-s + 863.·46-s + 155.·47-s + ⋯
L(s)  = 1  + 1.82·2-s + 2.32·4-s − 1.52·5-s + 2.41·8-s − 2.77·10-s − 0.831·11-s + 1.91·13-s + 2.07·16-s − 0.191·17-s + 0.0648·19-s − 3.53·20-s − 1.51·22-s + 1.51·23-s + 1.31·25-s + 3.48·26-s − 0.864·29-s + 0.109·31-s + 1.37·32-s − 0.348·34-s + 1.78·37-s + 0.118·38-s − 3.67·40-s + 1.65·41-s + 0.188·43-s − 1.93·44-s + 2.76·46-s + 0.481·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(78.0595\)
Root analytic conductor: \(8.83513\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{1323} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1323,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(5.769353722\)
\(L(\frac12)\) \(\approx\) \(5.769353722\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
good2 \( 1 - 5.15T + 8T^{2} \)
5 \( 1 + 17.0T + 125T^{2} \)
11 \( 1 + 30.3T + 1.33e3T^{2} \)
13 \( 1 - 89.5T + 2.19e3T^{2} \)
17 \( 1 + 13.4T + 4.91e3T^{2} \)
19 \( 1 - 5.37T + 6.85e3T^{2} \)
23 \( 1 - 167.T + 1.21e4T^{2} \)
29 \( 1 + 135.T + 2.43e4T^{2} \)
31 \( 1 - 18.9T + 2.97e4T^{2} \)
37 \( 1 - 402.T + 5.06e4T^{2} \)
41 \( 1 - 434.T + 6.89e4T^{2} \)
43 \( 1 - 53.1T + 7.95e4T^{2} \)
47 \( 1 - 155.T + 1.03e5T^{2} \)
53 \( 1 + 301.T + 1.48e5T^{2} \)
59 \( 1 + 412.T + 2.05e5T^{2} \)
61 \( 1 - 571.T + 2.26e5T^{2} \)
67 \( 1 - 820.T + 3.00e5T^{2} \)
71 \( 1 + 8.95T + 3.57e5T^{2} \)
73 \( 1 + 21.9T + 3.89e5T^{2} \)
79 \( 1 - 619.T + 4.93e5T^{2} \)
83 \( 1 - 259.T + 5.71e5T^{2} \)
89 \( 1 - 484.T + 7.04e5T^{2} \)
97 \( 1 - 252.T + 9.12e5T^{2} \)
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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

−9.113912978028741830044018854135, −8.047430028595814222394434173126, −7.50174896149147874827368716344, −6.56218254597625792967669205816, −5.77152848922439391695419823672, −4.83907158347323345339405052832, −4.03628000803736694765882855051, −3.47786240323766936653072639440, −2.58520971911799224825745820198, −0.941349331093177547140292894585, 0.941349331093177547140292894585, 2.58520971911799224825745820198, 3.47786240323766936653072639440, 4.03628000803736694765882855051, 4.83907158347323345339405052832, 5.77152848922439391695419823672, 6.56218254597625792967669205816, 7.50174896149147874827368716344, 8.047430028595814222394434173126, 9.113912978028741830044018854135

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