Properties

Label 2-1323-1.1-c1-0-31
Degree $2$
Conductor $1323$
Sign $-1$
Analytic cond. $10.5642$
Root an. cond. $3.25026$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.874·2-s − 1.23·4-s + 0.236·5-s + 2.82·8-s − 0.206·10-s − 0.540·11-s − 0.874·13-s − 5·17-s + 4.03·19-s − 0.291·20-s + 0.472·22-s − 5.99·23-s − 4.94·25-s + 0.763·26-s + 8.61·29-s + 6.53·31-s − 5.65·32-s + 4.37·34-s + 8.70·37-s − 3.52·38-s + 0.667·40-s − 8.70·41-s − 2.23·43-s + 0.667·44-s + 5.23·46-s − 7.47·47-s + 4.32·50-s + ⋯
L(s)  = 1  − 0.618·2-s − 0.618·4-s + 0.105·5-s + 0.999·8-s − 0.0652·10-s − 0.162·11-s − 0.242·13-s − 1.21·17-s + 0.925·19-s − 0.0652·20-s + 0.100·22-s − 1.24·23-s − 0.988·25-s + 0.149·26-s + 1.59·29-s + 1.17·31-s − 0.999·32-s + 0.749·34-s + 1.43·37-s − 0.572·38-s + 0.105·40-s − 1.35·41-s − 0.340·43-s + 0.100·44-s + 0.772·46-s − 1.08·47-s + 0.611·50-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(10.5642\)
Root analytic conductor: \(3.25026\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1323} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1323,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.874T + 2T^{2} \)
5 \( 1 - 0.236T + 5T^{2} \)
11 \( 1 + 0.540T + 11T^{2} \)
13 \( 1 + 0.874T + 13T^{2} \)
17 \( 1 + 5T + 17T^{2} \)
19 \( 1 - 4.03T + 19T^{2} \)
23 \( 1 + 5.99T + 23T^{2} \)
29 \( 1 - 8.61T + 29T^{2} \)
31 \( 1 - 6.53T + 31T^{2} \)
37 \( 1 - 8.70T + 37T^{2} \)
41 \( 1 + 8.70T + 41T^{2} \)
43 \( 1 + 2.23T + 43T^{2} \)
47 \( 1 + 7.47T + 47T^{2} \)
53 \( 1 + 3.16T + 53T^{2} \)
59 \( 1 + 13.9T + 59T^{2} \)
61 \( 1 + 0.540T + 61T^{2} \)
67 \( 1 + 6.76T + 67T^{2} \)
71 \( 1 - 6.73T + 71T^{2} \)
73 \( 1 + 13.3T + 73T^{2} \)
79 \( 1 + 2.52T + 79T^{2} \)
83 \( 1 + 4.23T + 83T^{2} \)
89 \( 1 + 11.7T + 89T^{2} \)
97 \( 1 + 5.11T + 97T^{2} \)
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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

−9.360819997364189504061994335964, −8.291782218709843934074080290246, −7.978358572593796618174358594342, −6.85815223054740220300002676493, −5.95606302773223515319506685285, −4.81786313234275237617708244101, −4.22620981602061599362381339749, −2.86706676752173035794115106752, −1.52706886126793220429518408191, 0, 1.52706886126793220429518408191, 2.86706676752173035794115106752, 4.22620981602061599362381339749, 4.81786313234275237617708244101, 5.95606302773223515319506685285, 6.85815223054740220300002676493, 7.978358572593796618174358594342, 8.291782218709843934074080290246, 9.360819997364189504061994335964

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