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Properties

Label 2-1323-63.41-c1-0-23
Degree $2$
Conductor $1323$
Sign $0.987 + 0.158i$
Analytic cond. $10.5642$
Root an. cond. $3.25026$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.5 + 0.866i)2-s + (0.5 − 0.866i)4-s + (1.5 − 2.59i)5-s − 1.73i·8-s + 5.19i·10-s + (1.5 − 0.866i)11-s + (1.5 + 0.866i)13-s + (2.49 + 4.33i)16-s − 3·17-s + 5.19i·19-s + (−1.50 − 2.59i)20-s + (−1.5 + 2.59i)22-s + (4.5 + 2.59i)23-s + (−2 − 3.46i)25-s − 3·26-s + ⋯
L(s)  = 1  + (−1.06 + 0.612i)2-s + (0.250 − 0.433i)4-s + (0.670 − 1.16i)5-s − 0.612i·8-s + 1.64i·10-s + (0.452 − 0.261i)11-s + (0.416 + 0.240i)13-s + (0.624 + 1.08i)16-s − 0.727·17-s + 1.19i·19-s + (−0.335 − 0.580i)20-s + (−0.319 + 0.553i)22-s + (0.938 + 0.541i)23-s + (−0.400 − 0.692i)25-s − 0.588·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 + 0.158i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.987 + 0.158i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $0.987 + 0.158i$
Analytic conductor: \(10.5642\)
Root analytic conductor: \(3.25026\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1323} (881, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1323,\ (\ :1/2),\ 0.987 + 0.158i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.051850228\)
\(L(\frac12)\) \(\approx\) \(1.051850228\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.5 - 0.866i)T + (1 - 1.73i)T^{2} \)
5 \( 1 + (-1.5 + 2.59i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.5 + 0.866i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.5 - 0.866i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + 3T + 17T^{2} \)
19 \( 1 - 5.19iT - 19T^{2} \)
23 \( 1 + (-4.5 - 2.59i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4.5 + 2.59i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (3 + 1.73i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 7T + 37T^{2} \)
41 \( 1 + (-1.5 + 2.59i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.5 + 0.866i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 8.66iT - 53T^{2} \)
59 \( 1 + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-12 + 6.92i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2 + 3.46i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 3.46iT - 71T^{2} \)
73 \( 1 + 5.19iT - 73T^{2} \)
79 \( 1 + (4 + 6.92i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (7.5 + 12.9i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 3T + 89T^{2} \)
97 \( 1 + (-1.5 + 0.866i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.452343792465186527144318567497, −8.715630154397928085371836593257, −8.306552609302574962413431944206, −7.31308050027186436270258386943, −6.37804935225052383484084021853, −5.71219349024419316037135419274, −4.59573602895056667858747484345, −3.60755652207958569346407357934, −1.81921505335063145846772891630, −0.798811610972492510804057422554, 1.07148710939418932926165939560, 2.39512660561262032965554588682, 2.94488852275526794249830348716, 4.47294253263176121643558856221, 5.60072337566334986200317027605, 6.64056105573029073589598917437, 7.13965319030429974490050212771, 8.359132464722967223231435292206, 9.027047913070435534133398151140, 9.697778134602527532931990691281

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