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Properties

Label 2-1323-63.59-c1-0-2
Degree $2$
Conductor $1323$
Sign $-0.795 - 0.606i$
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  + (1.02 + 0.592i)2-s + (−0.296 − 0.514i)4-s − 2.83·5-s − 3.07i·8-s + (−2.91 − 1.68i)10-s − 0.157i·11-s + (3.41 + 1.97i)13-s + (1.23 − 2.13i)16-s + (−2.07 + 3.58i)17-s + (−5.48 + 3.16i)19-s + (0.842 + 1.45i)20-s + (0.0935 − 0.162i)22-s + 0.546i·23-s + 3.05·25-s + (2.33 + 4.04i)26-s + ⋯
L(s)  = 1  + (0.726 + 0.419i)2-s + (−0.148 − 0.257i)4-s − 1.26·5-s − 1.08i·8-s + (−0.921 − 0.532i)10-s − 0.0475i·11-s + (0.947 + 0.546i)13-s + (0.307 − 0.532i)16-s + (−0.502 + 0.870i)17-s + (−1.25 + 0.726i)19-s + (0.188 + 0.326i)20-s + (0.0199 − 0.0345i)22-s + 0.113i·23-s + 0.610·25-s + (0.458 + 0.794i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.795 - 0.606i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.795 - 0.606i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $-0.795 - 0.606i$
Analytic conductor: \(10.5642\)
Root analytic conductor: \(3.25026\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1323} (962, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1323,\ (\ :1/2),\ -0.795 - 0.606i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6627633408\)
\(L(\frac12)\) \(\approx\) \(0.6627633408\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.02 - 0.592i)T + (1 + 1.73i)T^{2} \)
5 \( 1 + 2.83T + 5T^{2} \)
11 \( 1 + 0.157iT - 11T^{2} \)
13 \( 1 + (-3.41 - 1.97i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (2.07 - 3.58i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (5.48 - 3.16i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 - 0.546iT - 23T^{2} \)
29 \( 1 + (4.02 - 2.32i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-0.112 + 0.0647i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.23 - 2.13i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.99 - 3.45i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.28 - 5.68i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (4.33 - 7.50i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (2.25 + 1.30i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.80 - 3.12i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.91 + 1.68i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.663 + 1.14i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 0.409iT - 71T^{2} \)
73 \( 1 + (13.0 + 7.50i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (2.16 - 3.74i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-3.22 - 5.58i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (2.52 + 4.37i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-2.18 + 1.26i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.999501586188593433982080026876, −8.946733377144139072617288033587, −8.282250481954845738394242110755, −7.43011509435113786704190460457, −6.41166383052949474353013140675, −5.97273968086995468546757464295, −4.62628959696803209391891973540, −4.11625656867138093493984773099, −3.39224410705681482231303924280, −1.52439397906601071258412321713, 0.21872607722923654901599293966, 2.28306764018306707317121606651, 3.36165457295951805449274395871, 4.04965246898140162657486832855, 4.73313372470175534191048625501, 5.76471432806506453102589003481, 6.94165442354465326854588066402, 7.74702578051594601135741473908, 8.515838848839872138282153240664, 9.025770552114271771761222728658

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